| L(s) = 1 | − 2·3-s + 4-s − 3·5-s + 9-s − 3·11-s − 2·12-s + 6·15-s + 16-s − 3·20-s − 12·23-s − 25-s + 4·27-s + 31-s + 6·33-s + 36-s − 14·37-s − 3·44-s − 3·45-s + 3·47-s − 2·48-s − 4·49-s + 9·55-s − 3·59-s + 6·60-s + 64-s + 19·67-s + 24·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.34·5-s + 1/3·9-s − 0.904·11-s − 0.577·12-s + 1.54·15-s + 1/4·16-s − 0.670·20-s − 2.50·23-s − 1/5·25-s + 0.769·27-s + 0.179·31-s + 1.04·33-s + 1/6·36-s − 2.30·37-s − 0.452·44-s − 0.447·45-s + 0.437·47-s − 0.288·48-s − 4/7·49-s + 1.21·55-s − 0.390·59-s + 0.774·60-s + 1/8·64-s + 2.32·67-s + 2.88·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13068 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13068 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.12989033404875715521073637070, −10.55175822976249333480944695547, −10.18186630083785051983049905204, −9.518320070857030637017653764421, −8.408457085788223919586151992858, −8.094096292974237439032455042163, −7.63073733790111056785655429931, −6.85104324868188880585244307484, −6.32568429098919561502078174125, −5.56090928796558883985390242262, −5.08505285812300499666609056047, −4.09319439742608197495825532465, −3.51399773774521873504668584188, −2.18474835211874651912908800670, 0,
2.18474835211874651912908800670, 3.51399773774521873504668584188, 4.09319439742608197495825532465, 5.08505285812300499666609056047, 5.56090928796558883985390242262, 6.32568429098919561502078174125, 6.85104324868188880585244307484, 7.63073733790111056785655429931, 8.094096292974237439032455042163, 8.408457085788223919586151992858, 9.518320070857030637017653764421, 10.18186630083785051983049905204, 10.55175822976249333480944695547, 11.12989033404875715521073637070
