| L(s) = 1 | + 2-s − 0.820·3-s + 4-s + 4.15·5-s − 0.820·6-s − 7-s + 8-s − 2.32·9-s + 4.15·10-s − 4.85·11-s − 0.820·12-s + 3.30·13-s − 14-s − 3.41·15-s + 16-s + 1.18·17-s − 2.32·18-s − 2.34·19-s + 4.15·20-s + 0.820·21-s − 4.85·22-s + 6.19·23-s − 0.820·24-s + 12.2·25-s + 3.30·26-s + 4.37·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.473·3-s + 0.5·4-s + 1.85·5-s − 0.335·6-s − 0.377·7-s + 0.353·8-s − 0.775·9-s + 1.31·10-s − 1.46·11-s − 0.236·12-s + 0.916·13-s − 0.267·14-s − 0.880·15-s + 0.250·16-s + 0.288·17-s − 0.548·18-s − 0.537·19-s + 0.929·20-s + 0.179·21-s − 1.03·22-s + 1.29·23-s − 0.167·24-s + 2.45·25-s + 0.647·26-s + 0.841·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.485639504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.485639504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 0.820T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 6.43T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 + 6.98T + 37T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 + 2.49T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 - 6.95T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 - 9.68T + 71T^{2} \) |
| 73 | \( 1 - 2.30T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 3.18T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.255197382059118650359288044358, −6.83777399418774100862230974654, −6.53901116649525723815495337139, −5.80527943363313654917823580616, −5.21480137940280841569911626563, −4.96507854332428845543211734521, −3.44366068075271243720209741042, −2.74977520770602032992812421025, −2.11254254138995922004752712128, −0.904290645203233953205733955646,
0.904290645203233953205733955646, 2.11254254138995922004752712128, 2.74977520770602032992812421025, 3.44366068075271243720209741042, 4.96507854332428845543211734521, 5.21480137940280841569911626563, 5.80527943363313654917823580616, 6.53901116649525723815495337139, 6.83777399418774100862230974654, 8.255197382059118650359288044358
