| L(s) = 1 | + 2·2-s − 6.44·3-s + 4·4-s − 12.8·6-s + 8·8-s + 14.5·9-s + 48.3·11-s − 25.7·12-s − 93.4·13-s + 16·16-s + 20.2·17-s + 29.0·18-s + 31.0·19-s + 96.7·22-s + 21.0·23-s − 51.5·24-s − 186.·26-s + 80.5·27-s + 69.5·29-s − 161.·31-s + 32·32-s − 311.·33-s + 40.5·34-s + 58.0·36-s − 162.·37-s + 62.1·38-s + 602.·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.23·3-s + 0.5·4-s − 0.876·6-s + 0.353·8-s + 0.537·9-s + 1.32·11-s − 0.619·12-s − 1.99·13-s + 0.250·16-s + 0.289·17-s + 0.379·18-s + 0.375·19-s + 0.937·22-s + 0.190·23-s − 0.438·24-s − 1.41·26-s + 0.573·27-s + 0.445·29-s − 0.933·31-s + 0.176·32-s − 1.64·33-s + 0.204·34-s + 0.268·36-s − 0.722·37-s + 0.265·38-s + 2.47·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 6.44T + 27T^{2} \) |
| 11 | \( 1 - 48.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 93.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 21.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 365.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 468.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 587.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 536.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 625.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 123.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 210.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 141.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 117.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 61.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 436.T + 9.12e5T^{2} \) |
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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
−7.964926400296694171342220502492, −6.99049374058131207988444822745, −6.69008589498121261176440200596, −5.68017408620926379688762880886, −5.12255807354115984469838518248, −4.45265759236545599031762624693, −3.45589903980860487633241300854, −2.33433140612753863629069071322, −1.15133085520098247864513335854, 0,
1.15133085520098247864513335854, 2.33433140612753863629069071322, 3.45589903980860487633241300854, 4.45265759236545599031762624693, 5.12255807354115984469838518248, 5.68017408620926379688762880886, 6.69008589498121261176440200596, 6.99049374058131207988444822745, 7.964926400296694171342220502492
