| L(s) = 1 | − 3-s − 2·4-s + 1.39·5-s + 9-s − 1.89·11-s + 2·12-s + 3.67·13-s − 1.39·15-s + 4·16-s − 2·17-s − 1.60·19-s − 2.78·20-s + 0.609·23-s − 3.06·25-s − 27-s − 2.78·29-s + 1.50·31-s + 1.89·33-s − 2·36-s − 37-s − 3.67·39-s − 41-s − 3.71·43-s + 3.78·44-s + 1.39·45-s + 11.2·47-s − 4·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.622·5-s + 0.333·9-s − 0.570·11-s + 0.577·12-s + 1.01·13-s − 0.359·15-s + 16-s − 0.485·17-s − 0.369·19-s − 0.622·20-s + 0.127·23-s − 0.613·25-s − 0.192·27-s − 0.516·29-s + 0.269·31-s + 0.329·33-s − 0.333·36-s − 0.164·37-s − 0.588·39-s − 0.156·41-s − 0.566·43-s + 0.570·44-s + 0.207·45-s + 1.63·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 1.39T + 5T^{2} \) |
| 11 | \( 1 + 1.89T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 0.609T + 23T^{2} \) |
| 29 | \( 1 + 2.78T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 5.67T + 67T^{2} \) |
| 71 | \( 1 + 9.34T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 - 1.82T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 7.24T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| 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
−7.80417459598997465068005574161, −6.95061675969266775785980414949, −5.98116020008955187499438427970, −5.72082382544437866921316380236, −4.85726982539894124169091456327, −4.18523586899205603851906328021, −3.39808979162830408444577257341, −2.20135711510671317084357618805, −1.16326780040488763507288567066, 0,
1.16326780040488763507288567066, 2.20135711510671317084357618805, 3.39808979162830408444577257341, 4.18523586899205603851906328021, 4.85726982539894124169091456327, 5.72082382544437866921316380236, 5.98116020008955187499438427970, 6.95061675969266775785980414949, 7.80417459598997465068005574161
