| L(s) = 1 | − 2.44·2-s + 3.97·4-s + 1.74·5-s + 5.10·7-s − 4.84·8-s − 4.26·10-s + 5.64·11-s + 6.72·13-s − 12.4·14-s + 3.87·16-s + 0.427·17-s − 1.22·19-s + 6.94·20-s − 13.7·22-s − 23-s − 1.95·25-s − 16.4·26-s + 20.3·28-s − 29-s + 5.44·31-s + 0.200·32-s − 1.04·34-s + 8.90·35-s − 10.5·37-s + 3.00·38-s − 8.44·40-s − 6.52·41-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.98·4-s + 0.780·5-s + 1.92·7-s − 1.71·8-s − 1.34·10-s + 1.70·11-s + 1.86·13-s − 3.33·14-s + 0.969·16-s + 0.103·17-s − 0.281·19-s + 1.55·20-s − 2.94·22-s − 0.208·23-s − 0.390·25-s − 3.22·26-s + 3.83·28-s − 0.185·29-s + 0.978·31-s + 0.0355·32-s − 0.179·34-s + 1.50·35-s − 1.73·37-s + 0.487·38-s − 1.33·40-s − 1.01·41-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
| \(L(1)\) |
\(\approx\) |
\(1.753804431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.753804431\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 - 1.74T + 5T^{2} \) |
| 7 | \( 1 - 5.10T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 - 0.427T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 6.52T + 41T^{2} \) |
| 43 | \( 1 + 4.88T + 43T^{2} \) |
| 47 | \( 1 - 8.92T + 47T^{2} \) |
| 53 | \( 1 + 5.99T + 53T^{2} \) |
| 59 | \( 1 + 0.440T + 59T^{2} \) |
| 61 | \( 1 + 6.08T + 61T^{2} \) |
| 67 | \( 1 - 6.28T + 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.12T + 83T^{2} \) |
| 89 | \( 1 + 4.45T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
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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
−8.276126417428014686678125211325, −7.71146674621064449299197314162, −6.72405850319876218630869606410, −6.29615125797643929796528953509, −5.46173757134720501505560528461, −4.41223755338400491151631537644, −3.53691171971485034882115258666, −2.02319332088903278881580135680, −1.57567426523469680412117593451, −1.06153835741648835313369173592,
1.06153835741648835313369173592, 1.57567426523469680412117593451, 2.02319332088903278881580135680, 3.53691171971485034882115258666, 4.41223755338400491151631537644, 5.46173757134720501505560528461, 6.29615125797643929796528953509, 6.72405850319876218630869606410, 7.71146674621064449299197314162, 8.276126417428014686678125211325
