| L(s) = 1 | − 1.41·3-s − 4.82·7-s − 0.999·9-s + 4.82·11-s − 0.585·13-s − 2.82·17-s − 19-s + 6.82·21-s + 7.65·23-s + 5.65·27-s + 3.65·29-s + 6.82·31-s − 6.82·33-s + 0.585·37-s + 0.828·39-s + 0.828·41-s − 8.82·43-s + 0.828·47-s + 16.3·49-s + 4.00·51-s − 11.8·53-s + 1.41·57-s − 6.82·59-s − 5.65·61-s + 4.82·63-s + 9.89·67-s − 10.8·69-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 1.82·7-s − 0.333·9-s + 1.45·11-s − 0.162·13-s − 0.685·17-s − 0.229·19-s + 1.49·21-s + 1.59·23-s + 1.08·27-s + 0.679·29-s + 1.22·31-s − 1.18·33-s + 0.0963·37-s + 0.132·39-s + 0.129·41-s − 1.34·43-s + 0.120·47-s + 2.33·49-s + 0.560·51-s − 1.63·53-s + 0.187·57-s − 0.888·59-s − 0.724·61-s + 0.608·63-s + 1.20·67-s − 1.30·69-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 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 - 0.585T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−8.236238955486967627145873701615, −6.93900525007064562148059668708, −6.47652130978766870358374084828, −6.27783181745442113668461148884, −5.16334678234527788482999954828, −4.35203108401345806685501364869, −3.35178775759764765629588637127, −2.73188695642437320439387161454, −1.11662327206019462499518944674, 0,
1.11662327206019462499518944674, 2.73188695642437320439387161454, 3.35178775759764765629588637127, 4.35203108401345806685501364869, 5.16334678234527788482999954828, 6.27783181745442113668461148884, 6.47652130978766870358374084828, 6.93900525007064562148059668708, 8.236238955486967627145873701615
