| L(s) = 1 | + 1.36·2-s − 0.146·4-s + 3.87·5-s − 4.18·7-s − 2.92·8-s + 5.27·10-s + 11-s + 6.69·13-s − 5.70·14-s − 3.68·16-s − 2.46·17-s + 0.542·19-s − 0.568·20-s + 1.36·22-s − 2.08·23-s + 10.0·25-s + 9.11·26-s + 0.614·28-s + 2.48·29-s − 6.44·31-s + 0.828·32-s − 3.36·34-s − 16.2·35-s + 11.3·37-s + 0.739·38-s − 11.3·40-s + 12.1·41-s + ⋯ |
| L(s) = 1 | + 0.962·2-s − 0.0733·4-s + 1.73·5-s − 1.58·7-s − 1.03·8-s + 1.66·10-s + 0.301·11-s + 1.85·13-s − 1.52·14-s − 0.921·16-s − 0.598·17-s + 0.124·19-s − 0.127·20-s + 0.290·22-s − 0.434·23-s + 2.00·25-s + 1.78·26-s + 0.116·28-s + 0.460·29-s − 1.15·31-s + 0.146·32-s − 0.576·34-s − 2.74·35-s + 1.86·37-s + 0.119·38-s − 1.79·40-s + 1.89·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.520614770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.520614770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 13 | \( 1 - 6.69T + 13T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 19 | \( 1 - 0.542T + 19T^{2} \) |
| 23 | \( 1 + 2.08T + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 1.28T + 43T^{2} \) |
| 47 | \( 1 + 6.95T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 67 | \( 1 - 2.88T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 3.24T + 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 7.77T + 89T^{2} \) |
| 97 | \( 1 + 8.63T + 97T^{2} \) |
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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
−8.249579341720222278473344243105, −6.81199454619734359600805675101, −6.26686932070162162797604055522, −6.04551915597211749516785295801, −5.43614635812171139350279080395, −4.37435903005487639942815700989, −3.63536607272016525645851523853, −2.96975970527184458935395025910, −2.11770283094474123606434118157, −0.859216512291006579460731516236,
0.859216512291006579460731516236, 2.11770283094474123606434118157, 2.96975970527184458935395025910, 3.63536607272016525645851523853, 4.37435903005487639942815700989, 5.43614635812171139350279080395, 6.04551915597211749516785295801, 6.26686932070162162797604055522, 6.81199454619734359600805675101, 8.249579341720222278473344243105
