| L(s) = 1 | − 2·2-s + 4·4-s + 29.5·7-s − 8·8-s + 46.5·11-s − 92.0·13-s − 59.0·14-s + 16·16-s + 4.53·17-s + 87.5·19-s − 93.0·22-s − 160.·23-s + 184.·26-s + 118.·28-s − 241.·29-s − 2.68·31-s − 32·32-s − 9.07·34-s + 20.6·37-s − 175.·38-s − 501.·41-s − 294.·43-s + 186.·44-s + 321.·46-s − 478.·47-s + 529.·49-s − 368.·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.59·7-s − 0.353·8-s + 1.27·11-s − 1.96·13-s − 1.12·14-s + 0.250·16-s + 0.0647·17-s + 1.05·19-s − 0.901·22-s − 1.45·23-s + 1.38·26-s + 0.797·28-s − 1.54·29-s − 0.0155·31-s − 0.176·32-s − 0.0457·34-s + 0.0915·37-s − 0.747·38-s − 1.90·41-s − 1.04·43-s + 0.637·44-s + 1.03·46-s − 1.48·47-s + 1.54·49-s − 0.982·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 29.5T + 343T^{2} \) |
| 11 | \( 1 - 46.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 92.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.53T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.68T + 2.97e4T^{2} \) |
| 37 | \( 1 - 20.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 501.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 478.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 243.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 383.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 132.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 582.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 566.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 839.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 451.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 301.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 739.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.14e3T + 9.12e5T^{2} \) |
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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
−8.859797745448568815929264437701, −7.901227127375190353490832355096, −7.50683512551363547112645333195, −6.58907484964457345296216443501, −5.37521599676824526705981986623, −4.71419509191795424004510418563, −3.55059678575613104638555545817, −2.08684076418720859304538382248, −1.48571359651296381742040880978, 0,
1.48571359651296381742040880978, 2.08684076418720859304538382248, 3.55059678575613104638555545817, 4.71419509191795424004510418563, 5.37521599676824526705981986623, 6.58907484964457345296216443501, 7.50683512551363547112645333195, 7.901227127375190353490832355096, 8.859797745448568815929264437701
