| L(s) = 1 | + 2.13·2-s + 3.12·3-s + 2.57·4-s + 6.68·6-s + 0.919·7-s + 1.22·8-s + 6.77·9-s + 3.49·11-s + 8.04·12-s − 0.479·13-s + 1.96·14-s − 2.53·16-s − 1.48·17-s + 14.4·18-s + 4.77·19-s + 2.87·21-s + 7.48·22-s + 1.29·23-s + 3.82·24-s − 1.02·26-s + 11.8·27-s + 2.36·28-s − 5.29·29-s − 0.255·31-s − 7.85·32-s + 10.9·33-s − 3.17·34-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.80·3-s + 1.28·4-s + 2.72·6-s + 0.347·7-s + 0.432·8-s + 2.25·9-s + 1.05·11-s + 2.32·12-s − 0.132·13-s + 0.525·14-s − 0.632·16-s − 0.360·17-s + 3.41·18-s + 1.09·19-s + 0.627·21-s + 1.59·22-s + 0.270·23-s + 0.779·24-s − 0.200·26-s + 2.27·27-s + 0.446·28-s − 0.984·29-s − 0.0458·31-s − 1.38·32-s + 1.90·33-s − 0.544·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.20505569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.20505569\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 - 3.12T + 3T^{2} \) |
| 7 | \( 1 - 0.919T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 + 0.479T + 13T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 0.255T + 31T^{2} \) |
| 37 | \( 1 - 2.92T + 37T^{2} \) |
| 41 | \( 1 + 5.69T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 - 9.99T + 53T^{2} \) |
| 59 | \( 1 - 0.302T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 - 3.91T + 67T^{2} \) |
| 71 | \( 1 + 7.43T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 6.46T + 83T^{2} \) |
| 89 | \( 1 - 0.738T + 89T^{2} \) |
| 97 | \( 1 + 5.18T + 97T^{2} \) |
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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
−8.117914322782898583978214907652, −7.05800701654044093349184811859, −6.92293504178055582555981706631, −5.76432776480407595762322805142, −4.93349501213243870318270510063, −4.22820620584547715900131703274, −3.59380796882211129069781852257, −3.11508950192454937693063378271, −2.22074372505025781277771114868, −1.46590745040686583502532340192,
1.46590745040686583502532340192, 2.22074372505025781277771114868, 3.11508950192454937693063378271, 3.59380796882211129069781852257, 4.22820620584547715900131703274, 4.93349501213243870318270510063, 5.76432776480407595762322805142, 6.92293504178055582555981706631, 7.05800701654044093349184811859, 8.117914322782898583978214907652
