| L(s) = 1 | + 2.68·2-s + 3-s + 5.21·4-s + 5-s + 2.68·6-s − 3.93·7-s + 8.63·8-s + 9-s + 2.68·10-s + 5.48·11-s + 5.21·12-s − 2.32·13-s − 10.5·14-s + 15-s + 12.7·16-s + 0.459·17-s + 2.68·18-s + 5.80·19-s + 5.21·20-s − 3.93·21-s + 14.7·22-s − 3.85·23-s + 8.63·24-s + 25-s − 6.25·26-s + 27-s − 20.5·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.60·4-s + 0.447·5-s + 1.09·6-s − 1.48·7-s + 3.05·8-s + 0.333·9-s + 0.849·10-s + 1.65·11-s + 1.50·12-s − 0.646·13-s − 2.82·14-s + 0.258·15-s + 3.19·16-s + 0.111·17-s + 0.633·18-s + 1.33·19-s + 1.16·20-s − 0.858·21-s + 3.14·22-s − 0.803·23-s + 1.76·24-s + 0.200·25-s − 1.22·26-s + 0.192·27-s − 3.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.334372436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.334372436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 + 2.32T + 13T^{2} \) |
| 17 | \( 1 - 0.459T + 17T^{2} \) |
| 19 | \( 1 - 5.80T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 + 3.72T + 37T^{2} \) |
| 41 | \( 1 - 0.883T + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 + 3.41T + 53T^{2} \) |
| 59 | \( 1 - 4.17T + 59T^{2} \) |
| 61 | \( 1 + 0.602T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 6.85T + 71T^{2} \) |
| 73 | \( 1 - 1.24T + 73T^{2} \) |
| 79 | \( 1 + 9.95T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 0.622T + 89T^{2} \) |
| 97 | \( 1 + 0.965T + 97T^{2} \) |
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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
−7.61555527027121963998195155157, −7.06253195086314816870603544985, −6.41203302570857496312933133369, −6.00301505338133570159480042259, −5.16770490484339340613408017503, −4.24385495378421744741208782353, −3.64416921606413637982066554760, −3.08854799384893474703183859505, −2.34459898612598052868826431365, −1.30959328477803845967702341307,
1.30959328477803845967702341307, 2.34459898612598052868826431365, 3.08854799384893474703183859505, 3.64416921606413637982066554760, 4.24385495378421744741208782353, 5.16770490484339340613408017503, 6.00301505338133570159480042259, 6.41203302570857496312933133369, 7.06253195086314816870603544985, 7.61555527027121963998195155157
