L(s) = 1 | − 1.99·2-s + 1.97·4-s + 0.735·5-s − 3.83·7-s + 0.0540·8-s − 1.46·10-s + 0.205·11-s + 6.99·13-s + 7.64·14-s − 4.05·16-s + 3.70·17-s + 4.87·19-s + 1.45·20-s − 0.409·22-s + 23-s − 4.45·25-s − 13.9·26-s − 7.56·28-s − 29-s + 5.25·31-s + 7.97·32-s − 7.39·34-s − 2.81·35-s + 4.20·37-s − 9.72·38-s + 0.0396·40-s + 11.5·41-s + ⋯ |
L(s) = 1 | − 1.40·2-s + 0.986·4-s + 0.328·5-s − 1.44·7-s + 0.0190·8-s − 0.463·10-s + 0.0619·11-s + 1.94·13-s + 2.04·14-s − 1.01·16-s + 0.899·17-s + 1.11·19-s + 0.324·20-s − 0.0873·22-s + 0.208·23-s − 0.891·25-s − 2.73·26-s − 1.42·28-s − 0.185·29-s + 0.943·31-s + 1.40·32-s − 1.26·34-s − 0.476·35-s + 0.691·37-s − 1.57·38-s + 0.00627·40-s + 1.80·41-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
\(L(1)\) |
\(\approx\) |
\(0.9883731101\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9883731101\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.99T + 2T^{2} \) |
| 5 | \( 1 - 0.735T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 0.205T + 11T^{2} \) |
| 13 | \( 1 - 6.99T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 4.20T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 6.44T + 43T^{2} \) |
| 47 | \( 1 + 0.322T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 7.87T + 67T^{2} \) |
| 71 | \( 1 + 8.34T + 71T^{2} \) |
| 73 | \( 1 - 1.72T + 73T^{2} \) |
| 79 | \( 1 + 2.27T + 79T^{2} \) |
| 83 | \( 1 + 2.31T + 83T^{2} \) |
| 89 | \( 1 + 9.00T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.053681528631485939884103575908, −7.66704712668774374289774640400, −6.71875579596287226721330751870, −6.13989309687818376721329656251, −5.62019937368588979640341011604, −4.24239259083216645502692358687, −3.44570358876929217320468658606, −2.67851766122620221925306365229, −1.38914434244661769335048076733, −0.72792252221076310540109223725,
0.72792252221076310540109223725, 1.38914434244661769335048076733, 2.67851766122620221925306365229, 3.44570358876929217320468658606, 4.24239259083216645502692358687, 5.62019937368588979640341011604, 6.13989309687818376721329656251, 6.71875579596287226721330751870, 7.66704712668774374289774640400, 8.053681528631485939884103575908