| L(s) = 1 | − 1.85·2-s + 1.43·4-s − 5-s + 4.90·7-s + 1.05·8-s + 1.85·10-s + 11-s − 4.61·13-s − 9.08·14-s − 4.81·16-s − 3.76·17-s + 4.84·19-s − 1.43·20-s − 1.85·22-s + 0.860·23-s + 25-s + 8.54·26-s + 7.01·28-s + 10.3·29-s + 7.81·31-s + 6.80·32-s + 6.97·34-s − 4.90·35-s − 4.67·37-s − 8.97·38-s − 1.05·40-s − 2.97·41-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.715·4-s − 0.447·5-s + 1.85·7-s + 0.373·8-s + 0.585·10-s + 0.301·11-s − 1.27·13-s − 2.42·14-s − 1.20·16-s − 0.913·17-s + 1.11·19-s − 0.319·20-s − 0.394·22-s + 0.179·23-s + 0.200·25-s + 1.67·26-s + 1.32·28-s + 1.92·29-s + 1.40·31-s + 1.20·32-s + 1.19·34-s − 0.829·35-s − 0.768·37-s − 1.45·38-s − 0.166·40-s − 0.464·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7900892850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7900892850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 7 | \( 1 - 4.90T + 7T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 - 0.860T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 7.81T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 + 0.907T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4.26T + 67T^{2} \) |
| 71 | \( 1 - 5.13T + 71T^{2} \) |
| 73 | \( 1 + 4.61T + 73T^{2} \) |
| 79 | \( 1 + 0.843T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 8.54T + 97T^{2} \) |
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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
−10.76900140910892200234545965914, −10.05822882445436100257913873031, −8.975745734557210987126883376254, −8.309174718169988868478717698509, −7.62955082757483303126641867451, −6.86678679431246136779381990506, −5.01828025470229711006454113063, −4.46482881184923673252292304455, −2.39522899981451333800254076348, −1.05758285324307008203378166171,
1.05758285324307008203378166171, 2.39522899981451333800254076348, 4.46482881184923673252292304455, 5.01828025470229711006454113063, 6.86678679431246136779381990506, 7.62955082757483303126641867451, 8.309174718169988868478717698509, 8.975745734557210987126883376254, 10.05822882445436100257913873031, 10.76900140910892200234545965914
