| L(s) = 1 | + 1.62·3-s + 4.74·7-s − 0.367·9-s − 4.48·11-s − 0.843·13-s − 5.52·17-s − 19-s + 7.69·21-s − 0.779·23-s − 5.46·27-s − 10.6·29-s + 8.65·31-s − 7.26·33-s − 1.62·37-s − 1.36·39-s + 4.73·41-s − 9.67·43-s + 3.18·47-s + 15.5·49-s − 8.96·51-s + 6.17·53-s − 1.62·57-s − 11.6·59-s + 6.48·61-s − 1.74·63-s − 14.8·67-s − 1.26·69-s + ⋯ |
| L(s) = 1 | + 0.936·3-s + 1.79·7-s − 0.122·9-s − 1.35·11-s − 0.233·13-s − 1.33·17-s − 0.229·19-s + 1.67·21-s − 0.162·23-s − 1.05·27-s − 1.97·29-s + 1.55·31-s − 1.26·33-s − 0.266·37-s − 0.219·39-s + 0.739·41-s − 1.47·43-s + 0.464·47-s + 2.21·49-s − 1.25·51-s + 0.848·53-s − 0.214·57-s − 1.52·59-s + 0.829·61-s − 0.219·63-s − 1.81·67-s − 0.152·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.62T + 3T^{2} \) |
| 7 | \( 1 - 4.74T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 + 0.843T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 23 | \( 1 + 0.779T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 - 8.65T + 31T^{2} \) |
| 37 | \( 1 + 1.62T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 + 9.67T + 43T^{2} \) |
| 47 | \( 1 - 3.18T + 47T^{2} \) |
| 53 | \( 1 - 6.17T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 0.303T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 0.779T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 6.17T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70665769792905280274767750330, −7.19913705188252656092608145911, −6.01519519727919061491491305801, −5.30067961098554145281629489367, −4.66004042325239143075021227400, −4.02803937079290144714990112801, −2.87290905228708237691989778970, −2.27937614530118140330154004047, −1.63604345286323831206982208344, 0,
1.63604345286323831206982208344, 2.27937614530118140330154004047, 2.87290905228708237691989778970, 4.02803937079290144714990112801, 4.66004042325239143075021227400, 5.30067961098554145281629489367, 6.01519519727919061491491305801, 7.19913705188252656092608145911, 7.70665769792905280274767750330
