| L(s) = 1 | − 2·2-s + 4·4-s + 14·5-s − 7·7-s − 8·8-s − 28·10-s − 8·11-s + 13·13-s + 14·14-s + 16·16-s + 98·17-s − 28·19-s + 56·20-s + 16·22-s + 52·23-s + 71·25-s − 26·26-s − 28·28-s + 2·29-s − 168·31-s − 32·32-s − 196·34-s − 98·35-s − 146·37-s + 56·38-s − 112·40-s + 514·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.25·5-s − 0.377·7-s − 0.353·8-s − 0.885·10-s − 0.219·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.39·17-s − 0.338·19-s + 0.626·20-s + 0.155·22-s + 0.471·23-s + 0.567·25-s − 0.196·26-s − 0.188·28-s + 0.0128·29-s − 0.973·31-s − 0.176·32-s − 0.988·34-s − 0.473·35-s − 0.648·37-s + 0.239·38-s − 0.442·40-s + 1.95·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.995868681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995868681\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 17 | \( 1 - 98 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 52 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 T + p^{3} T^{2} \) |
| 31 | \( 1 + 168 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 514 T + p^{3} T^{2} \) |
| 43 | \( 1 + 236 T + p^{3} T^{2} \) |
| 47 | \( 1 - 216 T + p^{3} T^{2} \) |
| 53 | \( 1 - 66 T + p^{3} T^{2} \) |
| 59 | \( 1 - 84 T + p^{3} T^{2} \) |
| 61 | \( 1 - 446 T + p^{3} T^{2} \) |
| 67 | \( 1 - 292 T + p^{3} T^{2} \) |
| 71 | \( 1 + 100 T + p^{3} T^{2} \) |
| 73 | \( 1 - 450 T + p^{3} T^{2} \) |
| 79 | \( 1 - 392 T + p^{3} T^{2} \) |
| 83 | \( 1 - 292 T + p^{3} T^{2} \) |
| 89 | \( 1 - 402 T + p^{3} T^{2} \) |
| 97 | \( 1 - 314 T + p^{3} T^{2} \) |
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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
−9.245741056713409224985669468891, −8.331442106969207161307431081783, −7.46679223201471169745491509597, −6.62863872110179988754040457478, −5.81073670433650485795916438004, −5.26857254541560132784767044082, −3.75640048238353027834013112773, −2.72428031022713652651710315545, −1.79638813216971969242447805777, −0.77008412537397593343811959902,
0.77008412537397593343811959902, 1.79638813216971969242447805777, 2.72428031022713652651710315545, 3.75640048238353027834013112773, 5.26857254541560132784767044082, 5.81073670433650485795916438004, 6.62863872110179988754040457478, 7.46679223201471169745491509597, 8.331442106969207161307431081783, 9.245741056713409224985669468891
