| L(s) = 1 | + 276·3-s + 3.58e6·7-s − 1.42e7·9-s + 4.78e7·11-s − 2.47e8·13-s + 2.12e9·17-s − 1.07e9·19-s + 9.89e8·21-s − 2.49e10·23-s − 7.89e9·27-s − 1.65e11·29-s + 1.00e11·31-s + 1.31e10·33-s − 4.24e10·37-s − 6.83e10·39-s − 1.38e12·41-s + 1.16e12·43-s + 1.64e12·47-s + 8.10e12·49-s + 5.87e11·51-s + 4.46e12·53-s − 2.96e11·57-s − 2.87e13·59-s + 1.57e13·61-s − 5.11e13·63-s − 6.16e13·67-s − 6.89e12·69-s + ⋯ |
| L(s) = 1 | + 0.0728·3-s + 1.64·7-s − 0.994·9-s + 0.739·11-s − 1.09·13-s + 1.25·17-s − 0.275·19-s + 0.119·21-s − 1.52·23-s − 0.145·27-s − 1.77·29-s + 0.657·31-s + 0.0538·33-s − 0.0735·37-s − 0.0797·39-s − 1.11·41-s + 0.655·43-s + 0.473·47-s + 1.70·49-s + 0.0916·51-s + 0.522·53-s − 0.0201·57-s − 1.50·59-s + 0.640·61-s − 1.63·63-s − 1.24·67-s − 0.111·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 92 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 512248 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 47801700 T + p^{15} T^{2} \) |
| 13 | \( 1 + 19060382 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 2127682062 T + p^{15} T^{2} \) |
| 19 | \( 1 + 56571724 p T + p^{15} T^{2} \) |
| 23 | \( 1 + 24982896168 T + p^{15} T^{2} \) |
| 29 | \( 1 + 165099671946 T + p^{15} T^{2} \) |
| 31 | \( 1 - 100736332256 T + p^{15} T^{2} \) |
| 37 | \( 1 + 42490420334 T + p^{15} T^{2} \) |
| 41 | \( 1 + 1388779245414 T + p^{15} T^{2} \) |
| 43 | \( 1 - 1168783477180 T + p^{15} T^{2} \) |
| 47 | \( 1 - 1645655322672 T + p^{15} T^{2} \) |
| 53 | \( 1 - 4469627500578 T + p^{15} T^{2} \) |
| 59 | \( 1 + 28794808426572 T + p^{15} T^{2} \) |
| 61 | \( 1 - 15719941145942 T + p^{15} T^{2} \) |
| 67 | \( 1 + 61627103890604 T + p^{15} T^{2} \) |
| 71 | \( 1 + 66780412989192 T + p^{15} T^{2} \) |
| 73 | \( 1 - 57749646345094 T + p^{15} T^{2} \) |
| 79 | \( 1 - 198700138788272 T + p^{15} T^{2} \) |
| 83 | \( 1 - 113345193514212 T + p^{15} T^{2} \) |
| 89 | \( 1 + 48230883277974 T + p^{15} T^{2} \) |
| 97 | \( 1 + 95121696327074 T + p^{15} T^{2} \) |
| 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
−10.55956456129545349485509879315, −9.329691502225487031456395817502, −8.194107962523295161491671423735, −7.52317725495461856032855612944, −5.89749130381425239736349817480, −5.00106665764164638171819133229, −3.81937862730184893346852830878, −2.35042311153114318691838388040, −1.40845637579216147863499720598, 0,
1.40845637579216147863499720598, 2.35042311153114318691838388040, 3.81937862730184893346852830878, 5.00106665764164638171819133229, 5.89749130381425239736349817480, 7.52317725495461856032855612944, 8.194107962523295161491671423735, 9.329691502225487031456395817502, 10.55956456129545349485509879315
