| L(s) = 1 | − 0.664·3-s + 0.345·7-s − 2.55·9-s + 3.51·11-s + 3.22·13-s − 4.88·17-s + 19-s − 0.229·21-s − 7.62·23-s + 3.69·27-s − 1.73·29-s − 4.76·31-s − 2.33·33-s + 1.86·37-s − 2.14·39-s + 6.76·41-s − 0.606·43-s − 3.34·47-s − 6.88·49-s + 3.24·51-s + 0.107·53-s − 0.664·57-s + 12.3·59-s − 8.41·61-s − 0.884·63-s − 2.23·67-s + 5.06·69-s + ⋯ |
| L(s) = 1 | − 0.383·3-s + 0.130·7-s − 0.852·9-s + 1.06·11-s + 0.894·13-s − 1.18·17-s + 0.229·19-s − 0.0501·21-s − 1.59·23-s + 0.710·27-s − 0.323·29-s − 0.855·31-s − 0.406·33-s + 0.307·37-s − 0.343·39-s + 1.05·41-s − 0.0924·43-s − 0.488·47-s − 0.982·49-s + 0.454·51-s + 0.0147·53-s − 0.0880·57-s + 1.60·59-s − 1.07·61-s − 0.111·63-s − 0.273·67-s + 0.610·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.664T + 3T^{2} \) |
| 7 | \( 1 - 0.345T + 7T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 4.76T + 31T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 + 0.606T + 43T^{2} \) |
| 47 | \( 1 + 3.34T + 47T^{2} \) |
| 53 | \( 1 - 0.107T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 + 0.536T + 71T^{2} \) |
| 73 | \( 1 + 5.57T + 73T^{2} \) |
| 79 | \( 1 + 2.67T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.245098151402001525541520375792, −7.38258184101036106601574655264, −6.32538285869862024986811479092, −6.14835844746175876651844696368, −5.19472068512857797717119603548, −4.20519312046796208485193517784, −3.60768974297536016359389490388, −2.42444030761923594410837368434, −1.40249609117630693939587679186, 0,
1.40249609117630693939587679186, 2.42444030761923594410837368434, 3.60768974297536016359389490388, 4.20519312046796208485193517784, 5.19472068512857797717119603548, 6.14835844746175876651844696368, 6.32538285869862024986811479092, 7.38258184101036106601574655264, 8.245098151402001525541520375792
