| L(s) = 1 | + 5-s − 3.12·7-s + 4·11-s + 7.12·13-s + 1.12·17-s − 1.12·19-s + 5.12·23-s + 25-s + 2·29-s − 3.12·31-s − 3.12·35-s + 3.12·37-s − 6.24·41-s − 4·43-s − 5.12·47-s + 2.75·49-s + 12.2·53-s + 4·55-s + 10.2·59-s + 6·61-s + 7.12·65-s − 8·67-s − 10.2·71-s − 8.24·73-s − 12.4·77-s + 15.1·79-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.18·7-s + 1.20·11-s + 1.97·13-s + 0.272·17-s − 0.257·19-s + 1.06·23-s + 0.200·25-s + 0.371·29-s − 0.560·31-s − 0.527·35-s + 0.513·37-s − 0.975·41-s − 0.609·43-s − 0.747·47-s + 0.393·49-s + 1.68·53-s + 0.539·55-s + 1.33·59-s + 0.768·61-s + 0.883·65-s − 0.977·67-s − 1.21·71-s − 0.965·73-s − 1.42·77-s + 1.70·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.342420631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.342420631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 2.24T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−8.439914725970747038292246958927, −7.09911388697602465908604396890, −6.65054274666194798336827535162, −6.08610715828858184376044710782, −5.46412072991822693014752692089, −4.28061382099855004078045895753, −3.57033675440102626384612038426, −3.03038358013232828235457994536, −1.70823334303374909577272147217, −0.864942721513197752730434794887,
0.864942721513197752730434794887, 1.70823334303374909577272147217, 3.03038358013232828235457994536, 3.57033675440102626384612038426, 4.28061382099855004078045895753, 5.46412072991822693014752692089, 6.08610715828858184376044710782, 6.65054274666194798336827535162, 7.09911388697602465908604396890, 8.439914725970747038292246958927
