| L(s) = 1 | − 2.85·3-s + 0.454·5-s − 2.18·7-s + 5.14·9-s − 1.15·11-s + 6.03·13-s − 1.29·15-s + 0.0535·17-s − 2.08·19-s + 6.23·21-s + 3.22·23-s − 4.79·25-s − 6.11·27-s − 1.62·29-s − 2.01·31-s + 3.30·33-s − 0.993·35-s − 3.49·37-s − 17.2·39-s + 2.85·41-s + 2.78·43-s + 2.33·45-s − 9.03·47-s − 2.21·49-s − 0.152·51-s − 4.09·53-s − 0.526·55-s + ⋯ |
| L(s) = 1 | − 1.64·3-s + 0.203·5-s − 0.826·7-s + 1.71·9-s − 0.349·11-s + 1.67·13-s − 0.334·15-s + 0.0129·17-s − 0.479·19-s + 1.36·21-s + 0.673·23-s − 0.958·25-s − 1.17·27-s − 0.300·29-s − 0.362·31-s + 0.575·33-s − 0.167·35-s − 0.574·37-s − 2.75·39-s + 0.446·41-s + 0.423·43-s + 0.348·45-s − 1.31·47-s − 0.317·49-s − 0.0213·51-s − 0.561·53-s − 0.0710·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 - 0.454T + 5T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 6.03T + 13T^{2} \) |
| 17 | \( 1 - 0.0535T + 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 23 | \( 1 - 3.22T + 23T^{2} \) |
| 29 | \( 1 + 1.62T + 29T^{2} \) |
| 31 | \( 1 + 2.01T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 4.98T + 67T^{2} \) |
| 71 | \( 1 + 2.04T + 71T^{2} \) |
| 73 | \( 1 + 0.0937T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 7.96T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 3.00T + 97T^{2} \) |
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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
−9.755177231397171584846261350308, −8.882783838892403351601893149465, −7.70526913673475339706696160793, −6.59638052797373504250322108220, −6.14406679120445079920516499997, −5.45982564643818948949031620559, −4.38547837940620592267950246285, −3.30341807160592479765108755243, −1.47839765356320320507605595593, 0,
1.47839765356320320507605595593, 3.30341807160592479765108755243, 4.38547837940620592267950246285, 5.45982564643818948949031620559, 6.14406679120445079920516499997, 6.59638052797373504250322108220, 7.70526913673475339706696160793, 8.882783838892403351601893149465, 9.755177231397171584846261350308
