| L(s) = 1 | − 2.54·2-s + 0.626·3-s + 4.45·4-s − 0.217·5-s − 1.59·6-s − 0.803·7-s − 6.24·8-s − 2.60·9-s + 0.552·10-s + 11-s + 2.79·12-s + 0.170·13-s + 2.04·14-s − 0.136·15-s + 6.96·16-s − 1.26·17-s + 6.62·18-s + 6.74·19-s − 0.969·20-s − 0.503·21-s − 2.54·22-s + 8.79·23-s − 3.91·24-s − 4.95·25-s − 0.432·26-s − 3.51·27-s − 3.58·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.361·3-s + 2.22·4-s − 0.0972·5-s − 0.650·6-s − 0.303·7-s − 2.20·8-s − 0.869·9-s + 0.174·10-s + 0.301·11-s + 0.806·12-s + 0.0472·13-s + 0.545·14-s − 0.0351·15-s + 1.74·16-s − 0.307·17-s + 1.56·18-s + 1.54·19-s − 0.216·20-s − 0.109·21-s − 0.541·22-s + 1.83·23-s − 0.799·24-s − 0.990·25-s − 0.0849·26-s − 0.676·27-s − 0.676·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 - 0.626T + 3T^{2} \) |
| 5 | \( 1 + 0.217T + 5T^{2} \) |
| 7 | \( 1 + 0.803T + 7T^{2} \) |
| 13 | \( 1 - 0.170T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 - 8.79T + 23T^{2} \) |
| 31 | \( 1 - 1.57T + 31T^{2} \) |
| 37 | \( 1 - 1.10T + 37T^{2} \) |
| 41 | \( 1 + 0.226T + 41T^{2} \) |
| 43 | \( 1 + 6.11T + 43T^{2} \) |
| 47 | \( 1 - 4.77T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 0.258T + 61T^{2} \) |
| 67 | \( 1 + 6.23T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 6.08T + 73T^{2} \) |
| 79 | \( 1 + 1.45T + 79T^{2} \) |
| 83 | \( 1 + 9.36T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 1.95T + 97T^{2} \) |
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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
−7.57090853064315416154226785781, −7.03572409758881373600283944415, −6.27720740752709194505319069680, −5.61950909721077731162038702225, −4.61956414954577095059665490283, −3.18232823966526188512853014909, −3.04095409083034315576582125243, −1.90308226164946797323111370163, −1.05330369087999518848233110390, 0,
1.05330369087999518848233110390, 1.90308226164946797323111370163, 3.04095409083034315576582125243, 3.18232823966526188512853014909, 4.61956414954577095059665490283, 5.61950909721077731162038702225, 6.27720740752709194505319069680, 7.03572409758881373600283944415, 7.57090853064315416154226785781
