| L(s) = 1 | + 2.50·3-s + 0.708·5-s − 3.63·7-s + 3.25·9-s + 4.92·11-s − 0.194·13-s + 1.77·15-s + 1.76·17-s + 2.48·19-s − 9.08·21-s − 1.92·23-s − 4.49·25-s + 0.648·27-s − 6.35·29-s + 9.91·31-s + 12.3·33-s − 2.57·35-s + 9.89·37-s − 0.485·39-s + 11.4·41-s + 4.40·43-s + 2.30·45-s − 7.19·47-s + 6.18·49-s + 4.40·51-s + 5.85·53-s + 3.49·55-s + ⋯ |
| L(s) = 1 | + 1.44·3-s + 0.316·5-s − 1.37·7-s + 1.08·9-s + 1.48·11-s − 0.0538·13-s + 0.457·15-s + 0.427·17-s + 0.569·19-s − 1.98·21-s − 0.400·23-s − 0.899·25-s + 0.124·27-s − 1.17·29-s + 1.78·31-s + 2.14·33-s − 0.434·35-s + 1.62·37-s − 0.0777·39-s + 1.79·41-s + 0.672·43-s + 0.344·45-s − 1.04·47-s + 0.883·49-s + 0.617·51-s + 0.803·53-s + 0.470·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.332011633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.332011633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 0.708T + 5T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 + 0.194T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 6.35T + 29T^{2} \) |
| 31 | \( 1 - 9.91T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4.40T + 43T^{2} \) |
| 47 | \( 1 + 7.19T + 47T^{2} \) |
| 53 | \( 1 - 5.85T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 1.50T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 7.26T + 73T^{2} \) |
| 79 | \( 1 + 8.51T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 9.27T + 97T^{2} \) |
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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
−8.514610410835122508796971914944, −7.79782877994444946695041025962, −7.07440486337959071040946110756, −6.27635374611760983756548165794, −5.73230949320165108578672302703, −4.18996882128663582840701124143, −3.77519848869925735411308907661, −2.94589204636063705367070609366, −2.24093728647029474978651017322, −1.01169673710640905571420885499,
1.01169673710640905571420885499, 2.24093728647029474978651017322, 2.94589204636063705367070609366, 3.77519848869925735411308907661, 4.18996882128663582840701124143, 5.73230949320165108578672302703, 6.27635374611760983756548165794, 7.07440486337959071040946110756, 7.79782877994444946695041025962, 8.514610410835122508796971914944
