| L(s) = 1 | − 2·3-s − 26·7-s − 23·9-s + 28·11-s + 12·13-s − 64·17-s + 60·19-s + 52·21-s + 58·23-s + 100·27-s + 90·29-s + 128·31-s − 56·33-s + 236·37-s − 24·39-s + 242·41-s − 362·43-s − 226·47-s + 333·49-s + 128·51-s − 108·53-s − 120·57-s + 20·59-s + 542·61-s + 598·63-s + 434·67-s − 116·69-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 1.40·7-s − 0.851·9-s + 0.767·11-s + 0.256·13-s − 0.913·17-s + 0.724·19-s + 0.540·21-s + 0.525·23-s + 0.712·27-s + 0.576·29-s + 0.741·31-s − 0.295·33-s + 1.04·37-s − 0.0985·39-s + 0.921·41-s − 1.28·43-s − 0.701·47-s + 0.970·49-s + 0.351·51-s − 0.279·53-s − 0.278·57-s + 0.0441·59-s + 1.13·61-s + 1.19·63-s + 0.791·67-s − 0.202·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.152635686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152635686\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 64 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 236 T + p^{3} T^{2} \) |
| 41 | \( 1 - 242 T + p^{3} T^{2} \) |
| 43 | \( 1 + 362 T + p^{3} T^{2} \) |
| 47 | \( 1 + 226 T + p^{3} T^{2} \) |
| 53 | \( 1 + 108 T + p^{3} T^{2} \) |
| 59 | \( 1 - 20 T + p^{3} T^{2} \) |
| 61 | \( 1 - 542 T + p^{3} T^{2} \) |
| 67 | \( 1 - 434 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 73 | \( 1 - 632 T + p^{3} T^{2} \) |
| 79 | \( 1 - 720 T + p^{3} T^{2} \) |
| 83 | \( 1 - 478 T + p^{3} T^{2} \) |
| 89 | \( 1 + 490 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1456 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.99384244587925400599830176806, −9.816565172266707057540172373269, −9.185645826282686450801510495935, −8.181782643830396162926701350027, −6.66858873074558629658346091442, −6.34713158720880303990299397279, −5.10755513074204629174673699580, −3.72181200667750156606665190485, −2.69188977380543605078092619993, −0.69824173013547367662006809369,
0.69824173013547367662006809369, 2.69188977380543605078092619993, 3.72181200667750156606665190485, 5.10755513074204629174673699580, 6.34713158720880303990299397279, 6.66858873074558629658346091442, 8.181782643830396162926701350027, 9.185645826282686450801510495935, 9.816565172266707057540172373269, 10.99384244587925400599830176806
