| L(s) = 1 | + 2·2-s − 2·3-s + 4·4-s + 18·5-s − 4·6-s + 7·7-s + 8·8-s − 23·9-s + 36·10-s − 11·11-s − 8·12-s + 56·13-s + 14·14-s − 36·15-s + 16·16-s + 36·17-s − 46·18-s − 28·19-s + 72·20-s − 14·21-s − 22·22-s + 180·23-s − 16·24-s + 199·25-s + 112·26-s + 100·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.384·3-s + 1/2·4-s + 1.60·5-s − 0.272·6-s + 0.377·7-s + 0.353·8-s − 0.851·9-s + 1.13·10-s − 0.301·11-s − 0.192·12-s + 1.19·13-s + 0.267·14-s − 0.619·15-s + 1/4·16-s + 0.513·17-s − 0.602·18-s − 0.338·19-s + 0.804·20-s − 0.145·21-s − 0.213·22-s + 1.63·23-s − 0.136·24-s + 1.59·25-s + 0.844·26-s + 0.712·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.917577395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.917577395\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 - 36 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 334 T + p^{3} T^{2} \) |
| 37 | \( 1 - 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 444 T + p^{3} T^{2} \) |
| 43 | \( 1 + 316 T + p^{3} T^{2} \) |
| 47 | \( 1 + 402 T + p^{3} T^{2} \) |
| 53 | \( 1 + 486 T + p^{3} T^{2} \) |
| 59 | \( 1 + 282 T + p^{3} T^{2} \) |
| 61 | \( 1 - 380 T + p^{3} T^{2} \) |
| 67 | \( 1 - 176 T + p^{3} T^{2} \) |
| 71 | \( 1 + 324 T + p^{3} T^{2} \) |
| 73 | \( 1 - 800 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1144 T + p^{3} T^{2} \) |
| 83 | \( 1 - 468 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−12.85426914036039494247438562000, −11.33913859448930593525837927929, −10.79568861849419599514645525730, −9.534750594954137451114633023171, −8.398701841336842039676276828045, −6.69353219345719681575517160956, −5.76103708587324493429656248640, −5.09979767709534230915331312883, −3.12808284505645228516634854995, −1.61040963792843664155491799353,
1.61040963792843664155491799353, 3.12808284505645228516634854995, 5.09979767709534230915331312883, 5.76103708587324493429656248640, 6.69353219345719681575517160956, 8.398701841336842039676276828045, 9.534750594954137451114633023171, 10.79568861849419599514645525730, 11.33913859448930593525837927929, 12.85426914036039494247438562000
