| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 4·11-s − 2·13-s + 16-s + 6·17-s − 2·20-s + 4·22-s + 4·23-s − 25-s − 2·26-s − 2·29-s − 4·31-s + 32-s + 6·34-s − 10·37-s − 2·40-s + 10·41-s + 4·43-s + 4·44-s + 4·46-s + 4·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.316·40-s + 1.56·41-s + 0.609·43-s + 0.603·44-s + 0.589·46-s + 0.583·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.940935976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.940935976\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.84351324830482614195159853030, −7.19583741265591956235027568701, −6.73529275791267126360213503833, −5.70322294388590675424631045525, −5.20173444424979461085997861151, −4.21125984248492975559254303317, −3.72105155927604818995516413488, −3.06380200487442055779450117560, −1.89717432081133819255264343684, −0.816282240239440914761831766450,
0.816282240239440914761831766450, 1.89717432081133819255264343684, 3.06380200487442055779450117560, 3.72105155927604818995516413488, 4.21125984248492975559254303317, 5.20173444424979461085997861151, 5.70322294388590675424631045525, 6.73529275791267126360213503833, 7.19583741265591956235027568701, 7.84351324830482614195159853030
