| L(s) = 1 | − 3-s − 2·4-s + 2·5-s + 7-s + 9-s + 2·12-s − 13-s − 2·15-s + 4·16-s − 6·19-s − 4·20-s − 21-s − 25-s − 27-s − 2·28-s − 4·29-s + 6·31-s + 2·35-s − 2·36-s + 6·37-s + 39-s − 2·41-s − 8·43-s + 2·45-s + 7·47-s − 4·48-s + 49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.577·12-s − 0.277·13-s − 0.516·15-s + 16-s − 1.37·19-s − 0.894·20-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.377·28-s − 0.742·29-s + 1.07·31-s + 0.338·35-s − 1/3·36-s + 0.986·37-s + 0.160·39-s − 0.312·41-s − 1.21·43-s + 0.298·45-s + 1.02·47-s − 0.577·48-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.224261348363525584861095349535, −7.60216248984607352468086709380, −6.37880123681074789209390131766, −6.01769513603768006210918295002, −5.04200881499053364950272332672, −4.60474643363595389015450940257, −3.67973256323435761306976739106, −2.36720380637350926696727581707, −1.35842267726965233826731769574, 0,
1.35842267726965233826731769574, 2.36720380637350926696727581707, 3.67973256323435761306976739106, 4.60474643363595389015450940257, 5.04200881499053364950272332672, 6.01769513603768006210918295002, 6.37880123681074789209390131766, 7.60216248984607352468086709380, 8.224261348363525584861095349535
