L(s) = 1 | + 0.414·3-s + 3.82·5-s + 4·7-s − 2.82·9-s − 2.41·11-s − 4.65·13-s + 1.58·15-s + 3.65·17-s + 2·19-s + 1.65·21-s + 4.82·23-s + 9.65·25-s − 2.41·27-s − 29-s + 8.41·31-s − 0.999·33-s + 15.3·35-s + 1.65·37-s − 1.92·39-s − 9.65·41-s + 1.58·43-s − 10.8·45-s + 12.0·47-s + 9·49-s + 1.51·51-s + 7·53-s − 9.24·55-s + ⋯ |
L(s) = 1 | + 0.239·3-s + 1.71·5-s + 1.51·7-s − 0.942·9-s − 0.727·11-s − 1.29·13-s + 0.409·15-s + 0.886·17-s + 0.458·19-s + 0.361·21-s + 1.00·23-s + 1.93·25-s − 0.464·27-s − 0.185·29-s + 1.51·31-s − 0.174·33-s + 2.58·35-s + 0.272·37-s − 0.308·39-s − 1.50·41-s + 0.241·43-s − 1.61·45-s + 1.76·47-s + 1.28·49-s + 0.212·51-s + 0.961·53-s − 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.812718785\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.812718785\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 31 | \( 1 - 8.41T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 9.72T + 79T^{2} \) |
| 83 | \( 1 + 8.82T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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
−9.213089171669633775604384141684, −8.513517490632061353446266994029, −7.76088747109815006498136000088, −6.94130597379586743367547325431, −5.61643652125552832441500562852, −5.41100645941451209678674775160, −4.63960705437135719866842929440, −2.85525304227617879043861589331, −2.36117123974742074144565212456, −1.24461224359917649004126320102,
1.24461224359917649004126320102, 2.36117123974742074144565212456, 2.85525304227617879043861589331, 4.63960705437135719866842929440, 5.41100645941451209678674775160, 5.61643652125552832441500562852, 6.94130597379586743367547325431, 7.76088747109815006498136000088, 8.513517490632061353446266994029, 9.213089171669633775604384141684