| L(s) = 1 | − 2.62·2-s + 4.90·4-s + 4.03·5-s − 1.30·7-s − 7.64·8-s − 10.5·10-s − 3.92·11-s + 4.45·13-s + 3.43·14-s + 10.2·16-s − 5.05·17-s − 4.27·19-s + 19.7·20-s + 10.3·22-s − 23-s + 11.2·25-s − 11.7·26-s − 6.42·28-s − 29-s + 3.94·31-s − 11.7·32-s + 13.2·34-s − 5.27·35-s + 10.9·37-s + 11.2·38-s − 30.8·40-s + 10.8·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.45·4-s + 1.80·5-s − 0.494·7-s − 2.70·8-s − 3.34·10-s − 1.18·11-s + 1.23·13-s + 0.919·14-s + 2.56·16-s − 1.22·17-s − 0.980·19-s + 4.42·20-s + 2.19·22-s − 0.208·23-s + 2.24·25-s − 2.29·26-s − 1.21·28-s − 0.185·29-s + 0.709·31-s − 2.06·32-s + 2.27·34-s − 0.891·35-s + 1.79·37-s + 1.82·38-s − 4.86·40-s + 1.70·41-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
| \(L(1)\) |
\(\approx\) |
\(0.9705921804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9705921804\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 13 | \( 1 - 4.45T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 31 | \( 1 - 3.94T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 4.04T + 43T^{2} \) |
| 47 | \( 1 + 9.36T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 + 6.11T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 0.234T + 71T^{2} \) |
| 73 | \( 1 + 9.40T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 3.98T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.298278392564996270605275736713, −7.59298253697224751691128256484, −6.51942049416478815135751335915, −6.30579291006642066752009490801, −5.74802586253333679868224421106, −4.55395003346876898064594488005, −3.00434902389803854602670964202, −2.35481007752032398213495567458, −1.77103825424117766211859124981, −0.67346737824874749116825356008,
0.67346737824874749116825356008, 1.77103825424117766211859124981, 2.35481007752032398213495567458, 3.00434902389803854602670964202, 4.55395003346876898064594488005, 5.74802586253333679868224421106, 6.30579291006642066752009490801, 6.51942049416478815135751335915, 7.59298253697224751691128256484, 8.298278392564996270605275736713
