L(s) = 1 | + 0.852·2-s − 1.27·4-s + 0.278·5-s + 2.70·7-s − 2.79·8-s + 0.237·10-s − 5.07·11-s − 0.519·13-s + 2.30·14-s + 0.166·16-s − 2.19·17-s − 0.104·19-s − 0.354·20-s − 4.32·22-s − 23-s − 4.92·25-s − 0.443·26-s − 3.44·28-s − 29-s + 7.35·31-s + 5.72·32-s − 1.87·34-s + 0.752·35-s − 0.259·37-s − 0.0891·38-s − 0.776·40-s − 0.509·41-s + ⋯ |
L(s) = 1 | + 0.602·2-s − 0.636·4-s + 0.124·5-s + 1.02·7-s − 0.986·8-s + 0.0750·10-s − 1.52·11-s − 0.144·13-s + 0.616·14-s + 0.0415·16-s − 0.532·17-s − 0.0239·19-s − 0.0791·20-s − 0.921·22-s − 0.208·23-s − 0.984·25-s − 0.0869·26-s − 0.650·28-s − 0.185·29-s + 1.32·31-s + 1.01·32-s − 0.321·34-s + 0.127·35-s − 0.0426·37-s − 0.0144·38-s − 0.122·40-s − 0.0795·41-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
\(L(1)\) |
\(\approx\) |
\(1.823600067\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.823600067\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.852T + 2T^{2} \) |
| 5 | \( 1 - 0.278T + 5T^{2} \) |
| 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 + 5.07T + 11T^{2} \) |
| 13 | \( 1 + 0.519T + 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 + 0.104T + 19T^{2} \) |
| 31 | \( 1 - 7.35T + 31T^{2} \) |
| 37 | \( 1 + 0.259T + 37T^{2} \) |
| 41 | \( 1 + 0.509T + 41T^{2} \) |
| 43 | \( 1 - 4.63T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 + 0.323T + 59T^{2} \) |
| 61 | \( 1 - 8.63T + 61T^{2} \) |
| 67 | \( 1 + 3.31T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 0.479T + 73T^{2} \) |
| 79 | \( 1 - 9.38T + 79T^{2} \) |
| 83 | \( 1 - 6.67T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 - 0.587T + 97T^{2} \) |
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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
−8.041794645774545353414073687811, −7.58982854176219623981565457966, −6.48606237644065595629014813188, −5.64436399436920602484110547924, −5.17663961358017194438955733988, −4.52979364422728435619416991897, −3.87444821326226563724176333671, −2.76563603914740636929124704233, −2.09650578267924041782426858914, −0.63297655689971696667900003322,
0.63297655689971696667900003322, 2.09650578267924041782426858914, 2.76563603914740636929124704233, 3.87444821326226563724176333671, 4.52979364422728435619416991897, 5.17663961358017194438955733988, 5.64436399436920602484110547924, 6.48606237644065595629014813188, 7.58982854176219623981565457966, 8.041794645774545353414073687811