| L(s) = 1 | − 2.53·2-s + 0.274·3-s + 4.42·4-s + 5-s − 0.694·6-s + 7-s − 6.13·8-s − 2.92·9-s − 2.53·10-s + 1.69·11-s + 1.21·12-s − 2.53·14-s + 0.274·15-s + 6.70·16-s + 1.70·17-s + 7.41·18-s + 1.82·19-s + 4.42·20-s + 0.274·21-s − 4.29·22-s − 5.80·23-s − 1.68·24-s + 25-s − 1.62·27-s + 4.42·28-s − 4.92·29-s − 0.694·30-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.158·3-s + 2.21·4-s + 0.447·5-s − 0.283·6-s + 0.377·7-s − 2.16·8-s − 0.974·9-s − 0.801·10-s + 0.511·11-s + 0.349·12-s − 0.677·14-s + 0.0708·15-s + 1.67·16-s + 0.414·17-s + 1.74·18-s + 0.418·19-s + 0.988·20-s + 0.0598·21-s − 0.916·22-s − 1.21·23-s − 0.343·24-s + 0.200·25-s − 0.312·27-s + 0.835·28-s − 0.914·29-s − 0.126·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8890108759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8890108759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 - 0.274T + 3T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 + 5.80T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 8.41T + 31T^{2} \) |
| 37 | \( 1 + 0.501T + 37T^{2} \) |
| 41 | \( 1 - 1.49T + 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 - 9.47T + 47T^{2} \) |
| 53 | \( 1 + 2.95T + 53T^{2} \) |
| 59 | \( 1 + 0.143T + 59T^{2} \) |
| 61 | \( 1 - 5.71T + 61T^{2} \) |
| 67 | \( 1 + 7.51T + 67T^{2} \) |
| 71 | \( 1 - 2.65T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.274767620032139375855836681085, −7.62754148350234748635277848611, −6.98521727982747559118393710790, −6.02062983640987615394016986074, −5.71614851063240463995992282519, −4.41284673901110485243707541530, −3.23553937928328124697126469690, −2.41294740107538353486314711798, −1.65148349716207249439386668457, −0.65567228932255650066638465822,
0.65567228932255650066638465822, 1.65148349716207249439386668457, 2.41294740107538353486314711798, 3.23553937928328124697126469690, 4.41284673901110485243707541530, 5.71614851063240463995992282519, 6.02062983640987615394016986074, 6.98521727982747559118393710790, 7.62754148350234748635277848611, 8.274767620032139375855836681085
