| L(s) = 1 | − 0.301·2-s + 1.06·3-s − 1.90·4-s − 0.320·6-s + 2.50·7-s + 1.17·8-s − 1.87·9-s + 2.44·11-s − 2.02·12-s + 5.61·13-s − 0.757·14-s + 3.46·16-s + 0.565·18-s − 7.13·19-s + 2.66·21-s − 0.737·22-s + 0.860·23-s + 1.25·24-s − 1.69·26-s − 5.17·27-s − 4.79·28-s + 3.75·29-s − 2.24·31-s − 3.40·32-s + 2.59·33-s + 3.57·36-s + 5.10·37-s + ⋯ |
| L(s) = 1 | − 0.213·2-s + 0.612·3-s − 0.954·4-s − 0.130·6-s + 0.948·7-s + 0.416·8-s − 0.624·9-s + 0.737·11-s − 0.584·12-s + 1.55·13-s − 0.202·14-s + 0.865·16-s + 0.133·18-s − 1.63·19-s + 0.581·21-s − 0.157·22-s + 0.179·23-s + 0.255·24-s − 0.332·26-s − 0.995·27-s − 0.905·28-s + 0.696·29-s − 0.403·31-s − 0.601·32-s + 0.451·33-s + 0.596·36-s + 0.839·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.119175819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119175819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.301T + 2T^{2} \) |
| 3 | \( 1 - 1.06T + 3T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 19 | \( 1 + 7.13T + 19T^{2} \) |
| 23 | \( 1 - 0.860T + 23T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 - 2.30T + 47T^{2} \) |
| 53 | \( 1 + 2.77T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 - 0.906T + 61T^{2} \) |
| 67 | \( 1 + 6.69T + 67T^{2} \) |
| 71 | \( 1 + 0.240T + 71T^{2} \) |
| 73 | \( 1 - 6.45T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 0.395T + 89T^{2} \) |
| 97 | \( 1 - 8.95T + 97T^{2} \) |
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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
−8.161339585460356989892509327284, −7.57195697588470159396244699213, −6.36795119091568670927793186205, −5.90851471545645057747241927238, −4.97589969500098693138478315652, −4.11544143978302115583809029922, −3.83758909066826132073363286268, −2.69437280553148601119245413826, −1.68903042077313396766041899494, −0.78736538511145422949606366253,
0.78736538511145422949606366253, 1.68903042077313396766041899494, 2.69437280553148601119245413826, 3.83758909066826132073363286268, 4.11544143978302115583809029922, 4.97589969500098693138478315652, 5.90851471545645057747241927238, 6.36795119091568670927793186205, 7.57195697588470159396244699213, 8.161339585460356989892509327284
