| L(s) = 1 | + 2.65·5-s + 4.61·7-s − 4.21·11-s − 0.268·13-s − 0.800·17-s + 5.45·19-s + 4.25·23-s + 2.04·25-s + 8.09·29-s + 2.42·31-s + 12.2·35-s − 12.1·37-s − 4.26·41-s + 10.7·43-s − 6.40·47-s + 14.2·49-s − 4.79·53-s − 11.1·55-s − 0.639·59-s + 8.58·61-s − 0.713·65-s + 4.90·67-s − 15.9·71-s − 9.30·73-s − 19.4·77-s − 4.29·79-s + 13.9·83-s + ⋯ |
| L(s) = 1 | + 1.18·5-s + 1.74·7-s − 1.27·11-s − 0.0745·13-s − 0.194·17-s + 1.25·19-s + 0.887·23-s + 0.408·25-s + 1.50·29-s + 0.435·31-s + 2.06·35-s − 1.99·37-s − 0.666·41-s + 1.63·43-s − 0.934·47-s + 2.03·49-s − 0.658·53-s − 1.50·55-s − 0.0832·59-s + 1.09·61-s − 0.0884·65-s + 0.598·67-s − 1.88·71-s − 1.08·73-s − 2.21·77-s − 0.482·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.635748232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.635748232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.65T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 + 4.21T + 11T^{2} \) |
| 13 | \( 1 + 0.268T + 13T^{2} \) |
| 17 | \( 1 + 0.800T + 17T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 - 8.09T + 29T^{2} \) |
| 31 | \( 1 - 2.42T + 31T^{2} \) |
| 37 | \( 1 + 12.1T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 6.40T + 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 + 0.639T + 59T^{2} \) |
| 61 | \( 1 - 8.58T + 61T^{2} \) |
| 67 | \( 1 - 4.90T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 9.30T + 73T^{2} \) |
| 79 | \( 1 + 4.29T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 7.58T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−9.107709416804917439606087623266, −8.406077152365479761237789700107, −7.70377877894328814877489065304, −6.91435123792709989889498133223, −5.73875963014296019948356442694, −5.14154915051064405665947518770, −4.66802184376631729456866099440, −3.04796781204592831273844400755, −2.13960763436185910229564732794, −1.21824677010579125071475430284,
1.21824677010579125071475430284, 2.13960763436185910229564732794, 3.04796781204592831273844400755, 4.66802184376631729456866099440, 5.14154915051064405665947518770, 5.73875963014296019948356442694, 6.91435123792709989889498133223, 7.70377877894328814877489065304, 8.406077152365479761237789700107, 9.107709416804917439606087623266
