| L(s) = 1 | + 2.23·2-s + 3.00·4-s + 0.453·5-s + 2.23·7-s + 2.24·8-s + 1.01·10-s + 2.98·11-s + 0.660·13-s + 5.00·14-s − 0.991·16-s + 2.90·17-s + 5.95·19-s + 1.36·20-s + 6.67·22-s + 23-s − 4.79·25-s + 1.47·26-s + 6.71·28-s − 29-s + 1.21·31-s − 6.69·32-s + 6.50·34-s + 1.01·35-s − 2.58·37-s + 13.3·38-s + 1.01·40-s + 4.93·41-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.50·4-s + 0.202·5-s + 0.845·7-s + 0.792·8-s + 0.320·10-s + 0.899·11-s + 0.183·13-s + 1.33·14-s − 0.247·16-s + 0.705·17-s + 1.36·19-s + 0.304·20-s + 1.42·22-s + 0.208·23-s − 0.958·25-s + 0.289·26-s + 1.26·28-s − 0.185·29-s + 0.217·31-s − 1.18·32-s + 1.11·34-s + 0.171·35-s − 0.425·37-s + 2.16·38-s + 0.160·40-s + 0.770·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.573148913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.573148913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 0.453T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 - 0.660T + 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 - 4.83T + 43T^{2} \) |
| 47 | \( 1 + 3.25T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 9.01T + 59T^{2} \) |
| 61 | \( 1 + 6.99T + 61T^{2} \) |
| 67 | \( 1 - 5.10T + 67T^{2} \) |
| 71 | \( 1 + 6.78T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 - 9.99T + 79T^{2} \) |
| 83 | \( 1 + 9.24T + 83T^{2} \) |
| 89 | \( 1 - 6.86T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.72633418651053652926174546669, −7.30770843187750849106709288058, −6.32393537155699851787114381039, −5.79270511509002626077158762852, −5.15796534023898995984306372936, −4.48483885003609548413191083610, −3.73852269459337224835890513297, −3.10779127456057284456416878021, −2.06257789169422788626633670070, −1.17198921112745176671413277504,
1.17198921112745176671413277504, 2.06257789169422788626633670070, 3.10779127456057284456416878021, 3.73852269459337224835890513297, 4.48483885003609548413191083610, 5.15796534023898995984306372936, 5.79270511509002626077158762852, 6.32393537155699851787114381039, 7.30770843187750849106709288058, 7.72633418651053652926174546669
