| L(s) = 1 | − 3.22·3-s + 2.43·5-s − 2.42·7-s + 7.41·9-s + 4.13·11-s − 6.00·13-s − 7.86·15-s + 2.79·17-s − 19-s + 7.83·21-s + 0.561·23-s + 0.935·25-s − 14.2·27-s + 6.38·29-s − 2.28·31-s − 13.3·33-s − 5.91·35-s + 8.62·37-s + 19.3·39-s − 5.14·41-s + 0.407·43-s + 18.0·45-s + 1.84·47-s − 1.11·49-s − 9.02·51-s + 1.53·53-s + 10.0·55-s + ⋯ |
| L(s) = 1 | − 1.86·3-s + 1.08·5-s − 0.917·7-s + 2.47·9-s + 1.24·11-s − 1.66·13-s − 2.03·15-s + 0.678·17-s − 0.229·19-s + 1.70·21-s + 0.117·23-s + 0.187·25-s − 2.74·27-s + 1.18·29-s − 0.410·31-s − 2.32·33-s − 0.999·35-s + 1.41·37-s + 3.10·39-s − 0.802·41-s + 0.0620·43-s + 2.69·45-s + 0.268·47-s − 0.158·49-s − 1.26·51-s + 0.211·53-s + 1.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.009362893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.009362893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 3.22T + 3T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 - 4.13T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 23 | \( 1 - 0.561T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 - 8.62T + 37T^{2} \) |
| 41 | \( 1 + 5.14T + 41T^{2} \) |
| 43 | \( 1 - 0.407T + 43T^{2} \) |
| 47 | \( 1 - 1.84T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 - 2.20T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 9.26T + 67T^{2} \) |
| 71 | \( 1 + 2.66T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 - 7.49T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.82118330747537034867003608943, −6.82883434321916647960651602829, −6.66449162253850177131764752735, −5.94629654728962646360410367018, −5.38539341286982812379747144041, −4.70597917387470024583198325920, −3.89821458735288630029256108042, −2.64377759445916511491559422358, −1.57264245299024368138427760268, −0.60409910845735657706046026025,
0.60409910845735657706046026025, 1.57264245299024368138427760268, 2.64377759445916511491559422358, 3.89821458735288630029256108042, 4.70597917387470024583198325920, 5.38539341286982812379747144041, 5.94629654728962646360410367018, 6.66449162253850177131764752735, 6.82883434321916647960651602829, 7.82118330747537034867003608943
