| L(s) = 1 | + 5·2-s − 3·3-s + 17·4-s + 5·5-s − 15·6-s + 16·7-s + 45·8-s + 9·9-s + 25·10-s − 44·11-s − 51·12-s + 78·13-s + 80·14-s − 15·15-s + 89·16-s + 18·17-s + 45·18-s − 28·19-s + 85·20-s − 48·21-s − 220·22-s + 184·23-s − 135·24-s + 25·25-s + 390·26-s − 27·27-s + 272·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.577·3-s + 17/8·4-s + 0.447·5-s − 1.02·6-s + 0.863·7-s + 1.98·8-s + 1/3·9-s + 0.790·10-s − 1.20·11-s − 1.22·12-s + 1.66·13-s + 1.52·14-s − 0.258·15-s + 1.39·16-s + 0.256·17-s + 0.589·18-s − 0.338·19-s + 0.950·20-s − 0.498·21-s − 2.13·22-s + 1.66·23-s − 1.14·24-s + 1/5·25-s + 2.94·26-s − 0.192·27-s + 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.665892237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.665892237\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 29 | \( 1 - p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 78 T + p^{3} T^{2} \) |
| 43 | \( 1 + 260 T + p^{3} T^{2} \) |
| 47 | \( 1 - 312 T + p^{3} T^{2} \) |
| 53 | \( 1 - 574 T + p^{3} T^{2} \) |
| 59 | \( 1 - 180 T + p^{3} T^{2} \) |
| 61 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 340 T + p^{3} T^{2} \) |
| 71 | \( 1 - 296 T + p^{3} T^{2} \) |
| 73 | \( 1 - 394 T + p^{3} T^{2} \) |
| 79 | \( 1 + 960 T + p^{3} T^{2} \) |
| 83 | \( 1 + 908 T + p^{3} T^{2} \) |
| 89 | \( 1 + 990 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1234 T + p^{3} T^{2} \) |
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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
−11.01787903279084939908131226449, −10.47945589752012370275121669492, −8.792677476632474262964772608088, −7.58187155798137018964072446657, −6.54497752194227910628968504717, −5.59855281236004880130034921450, −5.11392487673221742565774919947, −4.03667569582335215129859164377, −2.80082205364819454527849557092, −1.44248852478742980714491400277,
1.44248852478742980714491400277, 2.80082205364819454527849557092, 4.03667569582335215129859164377, 5.11392487673221742565774919947, 5.59855281236004880130034921450, 6.54497752194227910628968504717, 7.58187155798137018964072446657, 8.792677476632474262964772608088, 10.47945589752012370275121669492, 11.01787903279084939908131226449
