| L(s) = 1 | − 5.41·2-s − 4.65·3-s + 21.3·4-s + 25.2·6-s − 72.0·8-s − 5.31·9-s − 52.2·11-s − 99.2·12-s + 30.6·13-s + 219.·16-s + 37.2·17-s + 28.7·18-s − 80.2·19-s + 282.·22-s − 25.8·23-s + 335.·24-s − 165.·26-s + 150.·27-s + 20.9·29-s + 314.·31-s − 613.·32-s + 243.·33-s − 201.·34-s − 113.·36-s − 197.·37-s + 434.·38-s − 142.·39-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.896·3-s + 2.66·4-s + 1.71·6-s − 3.18·8-s − 0.196·9-s − 1.43·11-s − 2.38·12-s + 0.654·13-s + 3.43·16-s + 0.531·17-s + 0.376·18-s − 0.968·19-s + 2.74·22-s − 0.234·23-s + 2.85·24-s − 1.25·26-s + 1.07·27-s + 0.134·29-s + 1.82·31-s − 3.38·32-s + 1.28·33-s − 1.01·34-s − 0.524·36-s − 0.875·37-s + 1.85·38-s − 0.586·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.41T + 8T^{2} \) |
| 3 | \( 1 + 4.65T + 27T^{2} \) |
| 11 | \( 1 + 52.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 11.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 33.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 153.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 616T + 2.05e5T^{2} \) |
| 61 | \( 1 + 15.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 166.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 952T + 3.57e5T^{2} \) |
| 73 | \( 1 + 148.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 660.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 45.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.68e3T + 9.12e5T^{2} \) |
| 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
−8.753121347542266478128498846534, −8.299694689327958777216252248981, −7.56593629459955471466369014225, −6.51287040492505255272973767710, −6.01292167170788064406480949044, −4.98641766015974544172455919274, −3.14854441715333259544207396108, −2.16547646323359179056678802407, −0.872637716312671756744814455202, 0,
0.872637716312671756744814455202, 2.16547646323359179056678802407, 3.14854441715333259544207396108, 4.98641766015974544172455919274, 6.01292167170788064406480949044, 6.51287040492505255272973767710, 7.56593629459955471466369014225, 8.299694689327958777216252248981, 8.753121347542266478128498846534
