| L(s) = 1 | − 2.40·5-s − 0.908·7-s + 5.49·13-s + 4.58·17-s − 4.40·19-s + 2.18·23-s + 0.779·25-s + 5.89·29-s − 4.40·31-s + 2.18·35-s − 4.99·37-s − 41-s − 4.18·43-s + 4.90·47-s − 6.17·49-s − 3.27·53-s + 11.0·59-s − 4.99·61-s − 13.2·65-s − 9.21·67-s − 8.30·71-s + 11.3·73-s + 1.71·79-s + 14.4·83-s − 11.0·85-s + 2.40·89-s − 4.99·91-s + ⋯ |
| L(s) = 1 | − 1.07·5-s − 0.343·7-s + 1.52·13-s + 1.11·17-s − 1.01·19-s + 0.455·23-s + 0.155·25-s + 1.09·29-s − 0.790·31-s + 0.369·35-s − 0.820·37-s − 0.156·41-s − 0.637·43-s + 0.715·47-s − 0.882·49-s − 0.449·53-s + 1.43·59-s − 0.639·61-s − 1.63·65-s − 1.12·67-s − 0.985·71-s + 1.33·73-s + 0.193·79-s + 1.58·83-s − 1.19·85-s + 0.254·89-s − 0.523·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.422498662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.422498662\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 + 0.908T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 - 4.58T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 - 2.18T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 + 4.40T + 31T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 + 3.27T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 4.99T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 8.30T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 1.71T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 2.40T + 89T^{2} \) |
| 97 | \( 1 + 0.624T + 97T^{2} \) |
| 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
−8.183202841124663818825054279083, −7.45435511723542138323952068286, −6.66577807446914351693342671609, −6.07202615284811647989716919311, −5.20140431057084054343978682720, −4.28748427618632602407553318217, −3.59925174329116305042909173940, −3.11220260266715805063586390701, −1.73797126314876198713196659305, −0.64098357939088040460010235263,
0.64098357939088040460010235263, 1.73797126314876198713196659305, 3.11220260266715805063586390701, 3.59925174329116305042909173940, 4.28748427618632602407553318217, 5.20140431057084054343978682720, 6.07202615284811647989716919311, 6.66577807446914351693342671609, 7.45435511723542138323952068286, 8.183202841124663818825054279083
