| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 5·7-s − 3·8-s + 9-s − 12-s + 4·13-s − 5·14-s − 16-s + 2·17-s + 18-s − 2·19-s − 5·21-s + 23-s − 3·24-s + 4·26-s + 27-s + 5·28-s − 5·29-s + 9·31-s + 5·32-s + 2·34-s − 36-s − 3·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.88·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s − 1.33·14-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.458·19-s − 1.09·21-s + 0.208·23-s − 0.612·24-s + 0.784·26-s + 0.192·27-s + 0.944·28-s − 0.928·29-s + 1.61·31-s + 0.883·32-s + 0.342·34-s − 1/6·36-s − 0.493·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.345763037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.345763037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.18336288760302, −12.83276259379291, −12.26156063588834, −11.86598280302984, −11.19559982354657, −10.63761689993856, −9.933353794906672, −9.760497569951884, −9.347680455014718, −8.786159945590415, −8.316651377941488, −8.003374881337302, −7.034377327390190, −6.661579093712023, −6.309102929815689, −5.694852207692145, −5.377134202759129, −4.454769446118330, −4.121942375548421, −3.466965064860873, −3.264761507112688, −2.780002550329106, −2.021478890331927, −1.052300077674996, −0.4151779225544627,
0.4151779225544627, 1.052300077674996, 2.021478890331927, 2.780002550329106, 3.264761507112688, 3.466965064860873, 4.121942375548421, 4.454769446118330, 5.377134202759129, 5.694852207692145, 6.309102929815689, 6.661579093712023, 7.034377327390190, 8.003374881337302, 8.316651377941488, 8.786159945590415, 9.347680455014718, 9.760497569951884, 9.933353794906672, 10.63761689993856, 11.19559982354657, 11.86598280302984, 12.26156063588834, 12.83276259379291, 13.18336288760302
