| L(s) = 1 | + 0.685·2-s − 1.53·4-s + 0.380·5-s − 0.405·7-s − 2.41·8-s + 0.261·10-s + 2.04·11-s + 4.25·13-s − 0.277·14-s + 1.40·16-s + 7.27·17-s − 2.41·19-s − 0.582·20-s + 1.40·22-s − 23-s − 4.85·25-s + 2.91·26-s + 0.620·28-s − 29-s − 4.43·31-s + 5.80·32-s + 4.98·34-s − 0.154·35-s + 0.231·37-s − 1.65·38-s − 0.921·40-s + 7.25·41-s + ⋯ |
| L(s) = 1 | + 0.484·2-s − 0.765·4-s + 0.170·5-s − 0.153·7-s − 0.855·8-s + 0.0825·10-s + 0.617·11-s + 1.18·13-s − 0.0742·14-s + 0.350·16-s + 1.76·17-s − 0.553·19-s − 0.130·20-s + 0.299·22-s − 0.208·23-s − 0.970·25-s + 0.572·26-s + 0.117·28-s − 0.185·29-s − 0.796·31-s + 1.02·32-s + 0.855·34-s − 0.0260·35-s + 0.0381·37-s − 0.268·38-s − 0.145·40-s + 1.13·41-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
| \(L(1)\) |
\(\approx\) |
\(2.180253074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180253074\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.685T + 2T^{2} \) |
| 5 | \( 1 - 0.380T + 5T^{2} \) |
| 7 | \( 1 + 0.405T + 7T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 17 | \( 1 - 7.27T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 - 0.231T + 37T^{2} \) |
| 41 | \( 1 - 7.25T + 41T^{2} \) |
| 43 | \( 1 + 0.152T + 43T^{2} \) |
| 47 | \( 1 - 6.36T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 2.00T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 2.73T + 71T^{2} \) |
| 73 | \( 1 + 6.65T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 + 7.16T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.126235616650965087952152118595, −7.46216098627004286630424573520, −6.32473026117719505663007109110, −5.90923459893857734401156100808, −5.30283201580840637542683205458, −4.29473902898219382723287443766, −3.71374034841329347656870058274, −3.14893319445074633130651915964, −1.78293752782965660662000653590, −0.75371198959746194983639070131,
0.75371198959746194983639070131, 1.78293752782965660662000653590, 3.14893319445074633130651915964, 3.71374034841329347656870058274, 4.29473902898219382723287443766, 5.30283201580840637542683205458, 5.90923459893857734401156100808, 6.32473026117719505663007109110, 7.46216098627004286630424573520, 8.126235616650965087952152118595
