| L(s) = 1 | + 1.40·2-s − 0.0314·4-s − 1.13·5-s − 1.90·7-s − 2.85·8-s − 1.58·10-s − 5.04·11-s + 3.78·13-s − 2.67·14-s − 3.93·16-s − 7.36·17-s − 0.713·19-s + 0.0356·20-s − 7.07·22-s + 23-s − 3.71·25-s + 5.30·26-s + 0.0600·28-s + 29-s + 4.34·31-s + 0.177·32-s − 10.3·34-s + 2.16·35-s + 7.55·37-s − 1.00·38-s + 3.22·40-s + 10.0·41-s + ⋯ |
| L(s) = 1 | + 0.992·2-s − 0.0157·4-s − 0.506·5-s − 0.721·7-s − 1.00·8-s − 0.502·10-s − 1.52·11-s + 1.04·13-s − 0.716·14-s − 0.984·16-s − 1.78·17-s − 0.163·19-s + 0.00796·20-s − 1.50·22-s + 0.208·23-s − 0.743·25-s + 1.04·26-s + 0.0113·28-s + 0.185·29-s + 0.779·31-s + 0.0314·32-s − 1.77·34-s + 0.365·35-s + 1.24·37-s − 0.162·38-s + 0.510·40-s + 1.56·41-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
| \(L(1)\) |
\(\approx\) |
\(1.183624925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183624925\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.40T + 2T^{2} \) |
| 5 | \( 1 + 1.13T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 5.04T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 + 7.36T + 17T^{2} \) |
| 19 | \( 1 + 0.713T + 19T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 - 7.55T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 - 1.16T + 47T^{2} \) |
| 53 | \( 1 - 4.59T + 53T^{2} \) |
| 59 | \( 1 + 4.46T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 0.947T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 9.66T + 89T^{2} \) |
| 97 | \( 1 + 3.19T + 97T^{2} \) |
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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
−8.213289787163431778046659440377, −7.23841861889088281710147892796, −6.34289354956026610512368784739, −6.01108451764882365312503583087, −5.03494760653985354323340382757, −4.43713858056634421782504547906, −3.76228905495490615699241846146, −2.97360442751050210781495936202, −2.28140332523013250940123959445, −0.46134227473056029607218521630,
0.46134227473056029607218521630, 2.28140332523013250940123959445, 2.97360442751050210781495936202, 3.76228905495490615699241846146, 4.43713858056634421782504547906, 5.03494760653985354323340382757, 6.01108451764882365312503583087, 6.34289354956026610512368784739, 7.23841861889088281710147892796, 8.213289787163431778046659440377
