| L(s) = 1 | + 3-s + 4.30·7-s + 9-s + 4.30·11-s + 4.30·13-s − 1.60·17-s − 2.60·19-s + 4.30·21-s + 23-s + 27-s − 2.60·29-s + 8.90·31-s + 4.30·33-s − 9.60·37-s + 4.30·39-s + 3·41-s − 6.21·43-s + 8.81·47-s + 11.5·49-s − 1.60·51-s + 10.3·53-s − 2.60·57-s + 14.1·59-s − 6.90·61-s + 4.30·63-s + 0.0916·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.62·7-s + 0.333·9-s + 1.29·11-s + 1.19·13-s − 0.389·17-s − 0.597·19-s + 0.938·21-s + 0.208·23-s + 0.192·27-s − 0.483·29-s + 1.59·31-s + 0.749·33-s − 1.57·37-s + 0.688·39-s + 0.468·41-s − 0.947·43-s + 1.28·47-s + 1.64·49-s − 0.224·51-s + 1.41·53-s − 0.345·57-s + 1.83·59-s − 0.884·61-s + 0.542·63-s + 0.0111·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.865085101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.865085101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 - 8.90T + 31T^{2} \) |
| 37 | \( 1 + 9.60T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 6.21T + 43T^{2} \) |
| 47 | \( 1 - 8.81T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 6.90T + 61T^{2} \) |
| 67 | \( 1 - 0.0916T + 67T^{2} \) |
| 71 | \( 1 + 2.90T + 71T^{2} \) |
| 73 | \( 1 + 1.78T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 1.78T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 9.51T + 97T^{2} \) |
| 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
−8.230187578769497228771782299623, −7.25664837583140521436913345121, −6.69707301709069272629618007899, −5.85667184372297883449301255426, −5.02502464166161654087178655210, −4.14295144130184860291123426666, −3.86009126763810245392147761324, −2.62040142805193559043825631547, −1.68838313494881635193804814166, −1.11980111159093447077078426548,
1.11980111159093447077078426548, 1.68838313494881635193804814166, 2.62040142805193559043825631547, 3.86009126763810245392147761324, 4.14295144130184860291123426666, 5.02502464166161654087178655210, 5.85667184372297883449301255426, 6.69707301709069272629618007899, 7.25664837583140521436913345121, 8.230187578769497228771782299623
