L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 3·9-s + 7·11-s + 2·13-s + 4·15-s − 6·17-s + 6·19-s + 2·21-s − 8·23-s + 3·25-s + 4·27-s + 6·29-s + 6·31-s + 14·33-s + 2·35-s + 37-s + 4·39-s + 12·41-s + 14·43-s + 6·45-s − 2·47-s − 9·49-s − 12·51-s − 2·53-s + 14·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 9-s + 2.11·11-s + 0.554·13-s + 1.03·15-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 1.66·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s + 1.07·31-s + 2.43·33-s + 0.338·35-s + 0.164·37-s + 0.640·39-s + 1.87·41-s + 2.13·43-s + 0.894·45-s − 0.291·47-s − 9/7·49-s − 1.68·51-s − 0.274·53-s + 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64641600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64641600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(10.86725402\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.86725402\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 19 T + 228 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_4$ | \( 1 + 23 T + 294 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 25 T + 346 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
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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
−8.101997652440054771685517138884, −7.60458727201949444606761689478, −7.27809998907248484671758255193, −7.19424823207499878875399265354, −6.33997071603033240948723384780, −6.33195227484538973456764689683, −6.06079224872589101788506472357, −5.92492918633722996570413706114, −4.88933334625498304352646673370, −4.87544408724898724780261989330, −4.29508896480258978866115539792, −4.21575450108730520272455516370, −3.51399730186506685658488625481, −3.51183954763214654532974670335, −2.62996698887298128459456258773, −2.60818400288396391195023370218, −1.91597852867178157556792825208, −1.69509706841606224362470294959, −1.06311274097880722048902186751, −0.807333485565959124916884558276,
0.807333485565959124916884558276, 1.06311274097880722048902186751, 1.69509706841606224362470294959, 1.91597852867178157556792825208, 2.60818400288396391195023370218, 2.62996698887298128459456258773, 3.51183954763214654532974670335, 3.51399730186506685658488625481, 4.21575450108730520272455516370, 4.29508896480258978866115539792, 4.87544408724898724780261989330, 4.88933334625498304352646673370, 5.92492918633722996570413706114, 6.06079224872589101788506472357, 6.33195227484538973456764689683, 6.33997071603033240948723384780, 7.19424823207499878875399265354, 7.27809998907248484671758255193, 7.60458727201949444606761689478, 8.101997652440054771685517138884