| L(s) = 1 | + 2.15·2-s + 1.75·3-s + 2.65·4-s + 3.78·6-s − 3.86·7-s + 1.40·8-s + 0.0781·9-s − 4.59·11-s + 4.65·12-s − 8.32·14-s − 2.26·16-s + 0.909·17-s + 0.168·18-s + 4.48·19-s − 6.77·21-s − 9.92·22-s − 5.76·23-s + 2.46·24-s − 5.12·27-s − 10.2·28-s − 1.52·29-s − 4.50·31-s − 7.70·32-s − 8.07·33-s + 1.96·34-s + 0.207·36-s + 3.73·37-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.01·3-s + 1.32·4-s + 1.54·6-s − 1.45·7-s + 0.497·8-s + 0.0260·9-s − 1.38·11-s + 1.34·12-s − 2.22·14-s − 0.567·16-s + 0.220·17-s + 0.0397·18-s + 1.02·19-s − 1.47·21-s − 2.11·22-s − 1.20·23-s + 0.503·24-s − 0.986·27-s − 1.93·28-s − 0.283·29-s − 0.808·31-s − 1.36·32-s − 1.40·33-s + 0.336·34-s + 0.0345·36-s + 0.613·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 - 1.75T + 3T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 17 | \( 1 - 0.909T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 + 4.50T + 31T^{2} \) |
| 37 | \( 1 - 3.73T + 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 - 4.80T + 43T^{2} \) |
| 47 | \( 1 + 4.37T + 47T^{2} \) |
| 53 | \( 1 - 2.58T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 5.30T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 6.95T + 79T^{2} \) |
| 83 | \( 1 - 1.81T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−7.80851739092909033758581824813, −7.29409217533169231628628452531, −6.28708419847689679857843018938, −5.72933380990462803037974420591, −5.07574966678161625233641904706, −3.95894217022720933730879798689, −3.36710230687488088520644126923, −2.85178297889531848398559480975, −2.16074366095528945959299157140, 0,
2.16074366095528945959299157140, 2.85178297889531848398559480975, 3.36710230687488088520644126923, 3.95894217022720933730879798689, 5.07574966678161625233641904706, 5.72933380990462803037974420591, 6.28708419847689679857843018938, 7.29409217533169231628628452531, 7.80851739092909033758581824813
