| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·6-s + 3·7-s + 2·8-s + 9-s − 3·11-s − 2·12-s − 13-s − 6·14-s − 4·16-s − 2·17-s − 2·18-s + 9·19-s − 6·21-s + 6·22-s + 3·23-s − 4·24-s + 2·26-s + 2·27-s + 3·28-s + 6·29-s + 16·31-s + 2·32-s + 6·33-s + 4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s + 1.13·7-s + 0.707·8-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 0.277·13-s − 1.60·14-s − 16-s − 0.485·17-s − 0.471·18-s + 2.06·19-s − 1.30·21-s + 1.27·22-s + 0.625·23-s − 0.816·24-s + 0.392·26-s + 0.384·27-s + 0.566·28-s + 1.11·29-s + 2.87·31-s + 0.353·32-s + 1.04·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7341166081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7341166081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 3 T - 3 T^{2} + 6 T^{3} + 32 T^{4} + 6 p T^{5} - 3 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - p T^{2} - 6 T^{3} + 180 T^{4} - 6 p T^{5} - p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 14 T^{2} - 32 T^{3} - 33 T^{4} - 32 p T^{5} - 14 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 9 T + 27 T^{2} - 144 T^{3} + 1016 T^{4} - 144 p T^{5} + 27 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - T^{2} + 108 T^{3} - 636 T^{4} + 108 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T - 14 T^{2} + 48 T^{3} + 615 T^{4} + 48 p T^{5} - 14 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3 T - 63 T^{2} + 6 T^{3} + 3482 T^{4} + 6 p T^{5} - 63 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 2 T - 62 T^{2} - 32 T^{3} + 2511 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T - 63 T^{2} + 10 T^{3} + 4820 T^{4} + 10 p T^{5} - 63 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 38 T^{2} + 256 T^{3} - 1509 T^{4} + 256 p T^{5} - 38 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 9 T - 23 T^{2} - 162 T^{3} + 2154 T^{4} - 162 p T^{5} - 23 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 3 T - 21 T^{2} - 312 T^{3} - 4192 T^{4} - 312 p T^{5} - 21 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 21 T + 193 T^{2} + 2226 T^{3} + 25152 T^{4} + 2226 p T^{5} + 193 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 + 11 T + 82 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 9 T + 182 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 4 T - 98 T^{2} - 256 T^{3} + 3651 T^{4} - 256 p T^{5} - 98 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 15 T + 81 T^{2} - 750 T^{3} - 10498 T^{4} - 750 p T^{5} + 81 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.80728338999127118504455037946, −6.21810027994020778675612429233, −6.13366966288620055430666944235, −5.91192646066611034894963075966, −5.62624651169231994252144084049, −5.60970730723455523042275500467, −5.15799545108976493374293148689, −5.04093014995318019956136220079, −4.78765305051872420537478003421, −4.76214168797417561251119486865, −4.51936593166882522589388401025, −4.37637041838467194253350379986, −3.88831806205572272985996982160, −3.80294833043411453821392761823, −3.28104430876495902267210323984, −3.02943362406411902535666802544, −2.87049496051891924484942461643, −2.54285196020158346336533423871, −2.38301992178887271462616337063, −2.02606642590935384560934506874, −1.47271067482213714227876390537, −1.24328915854516000488841986346, −1.03876680964278321437115073621, −0.75723357489625124904657591091, −0.29038981471890800377337932502,
0.29038981471890800377337932502, 0.75723357489625124904657591091, 1.03876680964278321437115073621, 1.24328915854516000488841986346, 1.47271067482213714227876390537, 2.02606642590935384560934506874, 2.38301992178887271462616337063, 2.54285196020158346336533423871, 2.87049496051891924484942461643, 3.02943362406411902535666802544, 3.28104430876495902267210323984, 3.80294833043411453821392761823, 3.88831806205572272985996982160, 4.37637041838467194253350379986, 4.51936593166882522589388401025, 4.76214168797417561251119486865, 4.78765305051872420537478003421, 5.04093014995318019956136220079, 5.15799545108976493374293148689, 5.60970730723455523042275500467, 5.62624651169231994252144084049, 5.91192646066611034894963075966, 6.13366966288620055430666944235, 6.21810027994020778675612429233, 6.80728338999127118504455037946
