| L(s) = 1 | − 2.82·5-s + 4.82·7-s − 4.82·11-s + 13-s + 4·17-s + 4.82·19-s − 4·23-s + 3.00·25-s + 4·29-s − 6.48·31-s − 13.6·35-s + 2·37-s + 6.82·41-s − 5.65·43-s + 10.4·47-s + 16.3·49-s + 13.6·53-s + 13.6·55-s + 4.82·59-s − 13.3·61-s − 2.82·65-s − 4.82·67-s + 2.48·71-s + 6·73-s − 23.3·77-s + 4·79-s + 3.17·83-s + ⋯ |
| L(s) = 1 | − 1.26·5-s + 1.82·7-s − 1.45·11-s + 0.277·13-s + 0.970·17-s + 1.10·19-s − 0.834·23-s + 0.600·25-s + 0.742·29-s − 1.16·31-s − 2.30·35-s + 0.328·37-s + 1.06·41-s − 0.862·43-s + 1.52·47-s + 2.33·49-s + 1.87·53-s + 1.84·55-s + 0.628·59-s − 1.70·61-s − 0.350·65-s − 0.589·67-s + 0.294·71-s + 0.702·73-s − 2.65·77-s + 0.450·79-s + 0.348·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.590018943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.590018943\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.82T + 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
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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
−8.982487848394555803249066795135, −8.173806635835078075053282353714, −7.63324601468895246082647633467, −7.46022914979616174454047540593, −5.74004936472521127431957991592, −5.13877191364845368954014183191, −4.34160834185350699106247809494, −3.43784561415096958347162701397, −2.23225466997827152293632114812, −0.876858085797912609153868185994,
0.876858085797912609153868185994, 2.23225466997827152293632114812, 3.43784561415096958347162701397, 4.34160834185350699106247809494, 5.13877191364845368954014183191, 5.74004936472521127431957991592, 7.46022914979616174454047540593, 7.63324601468895246082647633467, 8.173806635835078075053282353714, 8.982487848394555803249066795135
