| L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s + 13-s − 2·15-s + 17-s + 4·19-s − 25-s + 27-s − 2·29-s + 8·31-s − 4·33-s − 2·37-s + 39-s + 2·41-s + 4·43-s − 2·45-s − 8·47-s − 7·49-s + 51-s − 10·53-s + 8·55-s + 4·57-s − 4·59-s + 14·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.516·15-s + 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s − 49-s + 0.140·51-s − 1.37·53-s + 1.07·55-s + 0.529·57-s − 0.520·59-s + 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.61014588726586, −16.05807301178445, −15.66077655340819, −15.34131528858646, −14.50189349016622, −14.06228335148770, −13.36049626597087, −12.89202575254015, −12.28437361648148, −11.52596181852160, −11.20017647436948, −10.29410017376390, −9.872183175829691, −9.166664410080750, −8.314129172623915, −7.948822953231826, −7.530380404193677, −6.741207760872434, −5.910816029888285, −5.082467600945244, −4.512213701023178, −3.600395969072166, −3.108015208295201, −2.303139044636668, −1.179431515411365, 0,
1.179431515411365, 2.303139044636668, 3.108015208295201, 3.600395969072166, 4.512213701023178, 5.082467600945244, 5.910816029888285, 6.741207760872434, 7.530380404193677, 7.948822953231826, 8.314129172623915, 9.166664410080750, 9.872183175829691, 10.29410017376390, 11.20017647436948, 11.52596181852160, 12.28437361648148, 12.89202575254015, 13.36049626597087, 14.06228335148770, 14.50189349016622, 15.34131528858646, 15.66077655340819, 16.05807301178445, 16.61014588726586
