| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 4.23·5-s − 7-s + 2.23·8-s − 6.85·10-s + 6.23·13-s + 1.61·14-s − 4.85·16-s + 4.47·17-s + 3·19-s + 2.61·20-s − 8.47·23-s + 12.9·25-s − 10.0·26-s − 0.618·28-s + 3·29-s + 3.38·32-s − 7.23·34-s − 4.23·35-s + 3.47·37-s − 4.85·38-s + 9.47·40-s − 1.52·41-s + 10.9·43-s + 13.7·46-s + 3·47-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 1.89·5-s − 0.377·7-s + 0.790·8-s − 2.16·10-s + 1.72·13-s + 0.432·14-s − 1.21·16-s + 1.08·17-s + 0.688·19-s + 0.585·20-s − 1.76·23-s + 2.58·25-s − 1.97·26-s − 0.116·28-s + 0.557·29-s + 0.597·32-s − 1.24·34-s − 0.716·35-s + 0.570·37-s − 0.787·38-s + 1.49·40-s − 0.238·41-s + 1.66·43-s + 2.02·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.858428451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.858428451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 8.47T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.054889660924422808795874581962, −7.33614294437518082727250367668, −6.31375161914847456974448985304, −6.01770258814134310729353519639, −5.37132773687808546506323146494, −4.32029451758007965253316838943, −3.34848217830436975198088244269, −2.34574996071983003778520550714, −1.49227785884528830042038683493, −0.921392421691580142330643156260,
0.921392421691580142330643156260, 1.49227785884528830042038683493, 2.34574996071983003778520550714, 3.34848217830436975198088244269, 4.32029451758007965253316838943, 5.37132773687808546506323146494, 6.01770258814134310729353519639, 6.31375161914847456974448985304, 7.33614294437518082727250367668, 8.054889660924422808795874581962
