| L(s) = 1 | + 1.90·2-s + 3.07·3-s + 1.61·4-s + 5.85·6-s − 4.23·7-s − 0.726·8-s + 6.47·9-s + 2.23·11-s + 4.97·12-s − 6.15·13-s − 8.05·14-s − 4.61·16-s − 2.85·17-s + 12.3·18-s − 13.0·21-s + 4.25·22-s − 1.23·23-s − 2.23·24-s − 11.7·26-s + 10.6·27-s − 6.85·28-s − 7.60·29-s − 2.80·31-s − 7.33·32-s + 6.88·33-s − 5.42·34-s + 10.4·36-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 1.77·3-s + 0.809·4-s + 2.38·6-s − 1.60·7-s − 0.256·8-s + 2.15·9-s + 0.674·11-s + 1.43·12-s − 1.70·13-s − 2.15·14-s − 1.15·16-s − 0.692·17-s + 2.90·18-s − 2.84·21-s + 0.906·22-s − 0.257·23-s − 0.456·24-s − 2.29·26-s + 2.05·27-s − 1.29·28-s − 1.41·29-s − 0.502·31-s − 1.29·32-s + 1.19·33-s − 0.931·34-s + 1.74·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 3 | \( 1 - 3.07T + 3T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 6.15T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 - 1.90T + 37T^{2} \) |
| 41 | \( 1 + 2.62T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.61T + 61T^{2} \) |
| 67 | \( 1 - 4.53T + 67T^{2} \) |
| 71 | \( 1 + 7.77T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 7.38T + 83T^{2} \) |
| 89 | \( 1 - 5.60T + 89T^{2} \) |
| 97 | \( 1 - 8.05T + 97T^{2} \) |
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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
−7.17040029090339130166002769356, −6.82052998593051441494662865770, −6.02733785148984651732354891211, −5.14190931079467745423680893586, −4.16556723292816142755865095992, −3.89840415714369231413036163789, −3.08080650529063148457987605222, −2.63959545729758733568215287681, −1.93380127438494975636664453576, 0,
1.93380127438494975636664453576, 2.63959545729758733568215287681, 3.08080650529063148457987605222, 3.89840415714369231413036163789, 4.16556723292816142755865095992, 5.14190931079467745423680893586, 6.02733785148984651732354891211, 6.82052998593051441494662865770, 7.17040029090339130166002769356
