| L(s) = 1 | + 2.64·2-s − 3-s + 4.98·4-s + 5-s − 2.64·6-s + 1.94·7-s + 7.90·8-s + 9-s + 2.64·10-s − 4.54·11-s − 4.98·12-s + 1.94·13-s + 5.13·14-s − 15-s + 10.9·16-s + 2.55·17-s + 2.64·18-s − 3.99·19-s + 4.98·20-s − 1.94·21-s − 12.0·22-s − 0.669·23-s − 7.90·24-s + 25-s + 5.12·26-s − 27-s + 9.69·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.49·4-s + 0.447·5-s − 1.07·6-s + 0.734·7-s + 2.79·8-s + 0.333·9-s + 0.835·10-s − 1.37·11-s − 1.44·12-s + 0.538·13-s + 1.37·14-s − 0.258·15-s + 2.72·16-s + 0.620·17-s + 0.623·18-s − 0.916·19-s + 1.11·20-s − 0.424·21-s − 2.56·22-s − 0.139·23-s − 1.61·24-s + 0.200·25-s + 1.00·26-s − 0.192·27-s + 1.83·28-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
| \(L(1)\) |
\(\approx\) |
\(6.930478784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.930478784\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.64T + 2T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 11 | \( 1 + 4.54T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 + 3.99T + 19T^{2} \) |
| 23 | \( 1 + 0.669T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 - 5.69T + 31T^{2} \) |
| 37 | \( 1 - 7.57T + 37T^{2} \) |
| 41 | \( 1 - 8.46T + 41T^{2} \) |
| 43 | \( 1 - 0.745T + 43T^{2} \) |
| 47 | \( 1 - 2.25T + 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 + 0.383T + 59T^{2} \) |
| 61 | \( 1 + 4.26T + 61T^{2} \) |
| 67 | \( 1 - 7.20T + 67T^{2} \) |
| 71 | \( 1 + 0.721T + 71T^{2} \) |
| 73 | \( 1 + 0.889T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 + 6.65T + 83T^{2} \) |
| 89 | \( 1 - 7.35T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.81800603701063092261245448834, −7.10830470092216138591457158195, −6.12178605441887578791025615915, −5.94435341444370704046771367946, −5.08258555438082242530396619701, −4.64472670678456149091992512382, −3.92567523455910684807713318347, −2.80721338874250319569295833486, −2.27826396901332003127866612852, −1.14696940702074373511655660692,
1.14696940702074373511655660692, 2.27826396901332003127866612852, 2.80721338874250319569295833486, 3.92567523455910684807713318347, 4.64472670678456149091992512382, 5.08258555438082242530396619701, 5.94435341444370704046771367946, 6.12178605441887578791025615915, 7.10830470092216138591457158195, 7.81800603701063092261245448834
