| L(s) = 1 | − 2.61·2-s + 3-s + 4.85·4-s − 2.61·6-s − 1.38·7-s − 7.47·8-s + 9-s − 6.23·11-s + 4.85·12-s − 3·13-s + 3.61·14-s + 9.85·16-s − 3.38·17-s − 2.61·18-s − 19-s − 1.38·21-s + 16.3·22-s + 6.70·23-s − 7.47·24-s + 7.85·26-s + 27-s − 6.70·28-s − 4.23·29-s − 3.09·31-s − 10.8·32-s − 6.23·33-s + 8.85·34-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.42·4-s − 1.06·6-s − 0.522·7-s − 2.64·8-s + 0.333·9-s − 1.88·11-s + 1.40·12-s − 0.832·13-s + 0.966·14-s + 2.46·16-s − 0.820·17-s − 0.617·18-s − 0.229·19-s − 0.301·21-s + 3.48·22-s + 1.39·23-s − 1.52·24-s + 1.54·26-s + 0.192·27-s − 1.26·28-s − 0.786·29-s − 0.555·31-s − 1.91·32-s − 1.08·33-s + 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 6.23T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 3.38T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 - 2.38T + 43T^{2} \) |
| 47 | \( 1 + 9.18T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 1.14T + 71T^{2} \) |
| 73 | \( 1 + 9.14T + 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 + 5.47T + 89T^{2} \) |
| 97 | \( 1 + 9.56T + 97T^{2} \) |
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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
−10.55887211108849755876295886160, −9.880545747221439910647682331368, −9.074487389090352765113032849204, −8.267966413514602633451924170182, −7.43848006566903800052838530149, −6.75285382888558235975219297429, −5.20065415454096891198084754689, −3.04176700266513599153736635354, −2.12976083487618639891902727575, 0,
2.12976083487618639891902727575, 3.04176700266513599153736635354, 5.20065415454096891198084754689, 6.75285382888558235975219297429, 7.43848006566903800052838530149, 8.267966413514602633451924170182, 9.074487389090352765113032849204, 9.880545747221439910647682331368, 10.55887211108849755876295886160
