| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s − 6·11-s − 12-s + 4·13-s + 16-s − 2·18-s + 2·19-s − 6·22-s + 3·23-s − 24-s + 4·26-s + 5·27-s − 3·29-s + 8·31-s + 32-s + 6·33-s − 2·36-s + 4·37-s + 2·38-s − 4·39-s + 9·41-s + 7·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 1.80·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.471·18-s + 0.458·19-s − 1.27·22-s + 0.625·23-s − 0.204·24-s + 0.784·26-s + 0.962·27-s − 0.557·29-s + 1.43·31-s + 0.176·32-s + 1.04·33-s − 1/3·36-s + 0.657·37-s + 0.324·38-s − 0.640·39-s + 1.40·41-s + 1.06·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.015815221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015815221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.811623718879091610886654482206, −8.041671191522387879124626303592, −7.38329902349804359103319311042, −6.30031358215308733298575518492, −5.73485051529073623317717850528, −5.15165100271367887867751749250, −4.27516174634819019215803118177, −3.10646029962601689623238369280, −2.47698364576475141342970001572, −0.830562042269725958351888214521,
0.830562042269725958351888214521, 2.47698364576475141342970001572, 3.10646029962601689623238369280, 4.27516174634819019215803118177, 5.15165100271367887867751749250, 5.73485051529073623317717850528, 6.30031358215308733298575518492, 7.38329902349804359103319311042, 8.041671191522387879124626303592, 8.811623718879091610886654482206
