| L(s) = 1 | + 2.56·2-s − 3-s + 4.57·4-s + 5-s − 2.56·6-s − 3.28·7-s + 6.61·8-s + 9-s + 2.56·10-s + 0.413·11-s − 4.57·12-s + 4.70·13-s − 8.43·14-s − 15-s + 7.79·16-s − 4.34·17-s + 2.56·18-s + 4.72·19-s + 4.57·20-s + 3.28·21-s + 1.06·22-s + 1.54·23-s − 6.61·24-s + 25-s + 12.0·26-s − 27-s − 15.0·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 0.577·3-s + 2.28·4-s + 0.447·5-s − 1.04·6-s − 1.24·7-s + 2.33·8-s + 0.333·9-s + 0.811·10-s + 0.124·11-s − 1.32·12-s + 1.30·13-s − 2.25·14-s − 0.258·15-s + 1.94·16-s − 1.05·17-s + 0.604·18-s + 1.08·19-s + 1.02·20-s + 0.717·21-s + 0.226·22-s + 0.322·23-s − 1.34·24-s + 0.200·25-s + 2.36·26-s − 0.192·27-s − 2.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.619988553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.619988553\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 0.413T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 - 1.59T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 - 0.715T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 7.60T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 - 6.91T + 53T^{2} \) |
| 59 | \( 1 + 1.77T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 + 9.32T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 1.63T + 79T^{2} \) |
| 83 | \( 1 + 2.43T + 83T^{2} \) |
| 89 | \( 1 + 9.55T + 89T^{2} \) |
| 97 | \( 1 - 5.54T + 97T^{2} \) |
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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
−7.63573598709156700903802631947, −6.84813878815245305033752600619, −6.39807457725514527757068136099, −5.87628422449888601520221745048, −5.33640188071875133203953580350, −4.39538114126696825272253294593, −3.77692581349103479776193129100, −3.06194717395292906477515227363, −2.25179014746613875848197613377, −1.00286843625022467753633009254,
1.00286843625022467753633009254, 2.25179014746613875848197613377, 3.06194717395292906477515227363, 3.77692581349103479776193129100, 4.39538114126696825272253294593, 5.33640188071875133203953580350, 5.87628422449888601520221745048, 6.39807457725514527757068136099, 6.84813878815245305033752600619, 7.63573598709156700903802631947
