| L(s) = 1 | + 1.36·3-s − 4.25·5-s + 0.454·7-s − 1.14·9-s − 4.87·11-s − 3.14·13-s − 5.78·15-s − 2.81·17-s − 2.06·19-s + 0.618·21-s + 13.0·25-s − 5.64·27-s − 6.08·29-s + 8.93·31-s − 6.63·33-s − 1.93·35-s + 0.290·37-s − 4.28·39-s + 7.36·41-s + 9.73·43-s + 4.88·45-s + 7.00·47-s − 6.79·49-s − 3.82·51-s − 6.12·53-s + 20.7·55-s − 2.81·57-s + ⋯ |
| L(s) = 1 | + 0.785·3-s − 1.90·5-s + 0.171·7-s − 0.382·9-s − 1.47·11-s − 0.873·13-s − 1.49·15-s − 0.681·17-s − 0.474·19-s + 0.135·21-s + 2.61·25-s − 1.08·27-s − 1.13·29-s + 1.60·31-s − 1.15·33-s − 0.326·35-s + 0.0477·37-s − 0.686·39-s + 1.15·41-s + 1.48·43-s + 0.728·45-s + 1.02·47-s − 0.970·49-s − 0.535·51-s − 0.840·53-s + 2.79·55-s − 0.372·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7476656384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7476656384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 + 4.25T + 5T^{2} \) |
| 7 | \( 1 - 0.454T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 + 2.06T + 19T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 - 8.93T + 31T^{2} \) |
| 37 | \( 1 - 0.290T + 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 - 9.73T + 43T^{2} \) |
| 47 | \( 1 - 7.00T + 47T^{2} \) |
| 53 | \( 1 + 6.12T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 0.120T + 61T^{2} \) |
| 67 | \( 1 - 2.52T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 6.27T + 73T^{2} \) |
| 79 | \( 1 - 9.75T + 79T^{2} \) |
| 83 | \( 1 + 2.81T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + 4.20T + 97T^{2} \) |
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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
−8.155647045687231944042584850740, −7.78082443114008237904748078089, −7.40533046068600108155637251451, −6.32497911364934779467492848513, −5.16980304451432947172626546688, −4.51290812199582780159302907990, −3.80005712233441615439055683593, −2.86344088737261120061406244608, −2.37811382801305380513259787495, −0.43741995721352758800186104953,
0.43741995721352758800186104953, 2.37811382801305380513259787495, 2.86344088737261120061406244608, 3.80005712233441615439055683593, 4.51290812199582780159302907990, 5.16980304451432947172626546688, 6.32497911364934779467492848513, 7.40533046068600108155637251451, 7.78082443114008237904748078089, 8.155647045687231944042584850740
