L(s) = 1 | + 2.05·2-s + 2.20·4-s − 3.75·5-s − 2.35·7-s + 0.430·8-s − 7.69·10-s − 3.27·11-s + 4.29·13-s − 4.84·14-s − 3.53·16-s + 5.05·17-s − 7.69·19-s − 8.28·20-s − 6.71·22-s + 23-s + 9.07·25-s + 8.81·26-s − 5.21·28-s + 29-s − 8.24·31-s − 8.11·32-s + 10.3·34-s + 8.85·35-s − 7.63·37-s − 15.7·38-s − 1.61·40-s + 6.80·41-s + ⋯ |
L(s) = 1 | + 1.45·2-s + 1.10·4-s − 1.67·5-s − 0.891·7-s + 0.152·8-s − 2.43·10-s − 0.986·11-s + 1.19·13-s − 1.29·14-s − 0.884·16-s + 1.22·17-s − 1.76·19-s − 1.85·20-s − 1.43·22-s + 0.208·23-s + 1.81·25-s + 1.72·26-s − 0.985·28-s + 0.185·29-s − 1.48·31-s − 1.43·32-s + 1.77·34-s + 1.49·35-s − 1.25·37-s − 2.56·38-s − 0.255·40-s + 1.06·41-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
\(L(1)\) |
\(\approx\) |
\(1.863660591\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.863660591\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.05T + 2T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 7.69T + 19T^{2} \) |
| 31 | \( 1 + 8.24T + 31T^{2} \) |
| 37 | \( 1 + 7.63T + 37T^{2} \) |
| 41 | \( 1 - 6.80T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 8.31T + 53T^{2} \) |
| 59 | \( 1 - 6.64T + 59T^{2} \) |
| 61 | \( 1 + 4.78T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 - 8.12T + 97T^{2} \) |
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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
−7.980770823062418929435409141810, −7.16195930015222248047276213141, −6.59605401279776112983189566402, −5.73244207973470314808500784094, −5.20577924142763704729792136554, −4.07616682819754024632201320990, −3.85700285550375146797464498455, −3.22800026791352467723251386521, −2.35784776078108947060045585141, −0.54537786476120804254674032466,
0.54537786476120804254674032466, 2.35784776078108947060045585141, 3.22800026791352467723251386521, 3.85700285550375146797464498455, 4.07616682819754024632201320990, 5.20577924142763704729792136554, 5.73244207973470314808500784094, 6.59605401279776112983189566402, 7.16195930015222248047276213141, 7.980770823062418929435409141810