| L(s) = 1 | + 1.98·2-s − 3-s + 1.94·4-s − 3.52·5-s − 1.98·6-s − 0.111·8-s + 9-s − 7.00·10-s − 4.16·11-s − 1.94·12-s − 5.71·13-s + 3.52·15-s − 4.10·16-s − 3.10·17-s + 1.98·18-s + 2.60·19-s − 6.85·20-s − 8.27·22-s − 0.118·23-s + 0.111·24-s + 7.43·25-s − 11.3·26-s − 27-s + 5.45·29-s + 7.00·30-s − 8.12·31-s − 7.93·32-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 0.577·3-s + 0.972·4-s − 1.57·5-s − 0.810·6-s − 0.0392·8-s + 0.333·9-s − 2.21·10-s − 1.25·11-s − 0.561·12-s − 1.58·13-s + 0.910·15-s − 1.02·16-s − 0.753·17-s + 0.468·18-s + 0.596·19-s − 1.53·20-s − 1.76·22-s − 0.0246·23-s + 0.0226·24-s + 1.48·25-s − 2.22·26-s − 0.192·27-s + 1.01·29-s + 1.27·30-s − 1.46·31-s − 1.40·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6755972237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6755972237\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 + 3.10T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 + 0.118T + 23T^{2} \) |
| 29 | \( 1 - 5.45T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 - 9.28T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 2.57T + 67T^{2} \) |
| 71 | \( 1 - 2.06T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 - 8.80T + 79T^{2} \) |
| 83 | \( 1 - 2.26T + 83T^{2} \) |
| 89 | \( 1 - 9.72T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.71776178019846860109199987887, −7.03210306736605662563431034651, −6.57768696302657737025793855647, −5.30461892861175022491223445875, −5.11603536287608500580387190883, −4.48261452352846935951617035655, −3.73723567208627839326368071876, −3.01365022742536575978559954282, −2.22374085328209888672440412461, −0.31909326068888189698857626626,
0.31909326068888189698857626626, 2.22374085328209888672440412461, 3.01365022742536575978559954282, 3.73723567208627839326368071876, 4.48261452352846935951617035655, 5.11603536287608500580387190883, 5.30461892861175022491223445875, 6.57768696302657737025793855647, 7.03210306736605662563431034651, 7.71776178019846860109199987887
