| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 3·11-s − 2·13-s + 16-s + 3·17-s + 3·20-s − 3·22-s + 4·25-s + 2·26-s + 4·31-s − 32-s − 3·34-s + 5·37-s − 3·40-s − 3·41-s − 10·43-s + 3·44-s − 4·50-s − 2·52-s − 12·53-s + 9·55-s + 6·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.904·11-s − 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.670·20-s − 0.639·22-s + 4/5·25-s + 0.392·26-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.821·37-s − 0.474·40-s − 0.468·41-s − 1.52·43-s + 0.452·44-s − 0.565·50-s − 0.277·52-s − 1.64·53-s + 1.21·55-s + 0.781·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 318402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 318402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.89417356637478, −12.29336530498892, −11.95400238625713, −11.38732647745938, −11.08285859593699, −10.25031746221232, −10.05313537763523, −9.732814905684977, −9.316626042833336, −8.792690643437789, −8.392856158943386, −7.774785989556767, −7.357789469358524, −6.697327820630451, −6.357027069206774, −6.021451652152212, −5.354619817212085, −4.973649000674243, −4.318457941266468, −3.637801721853865, −2.959341779970725, −2.612412937109758, −1.766167524954445, −1.551315266744085, −0.8962493221830862, 0,
0.8962493221830862, 1.551315266744085, 1.766167524954445, 2.612412937109758, 2.959341779970725, 3.637801721853865, 4.318457941266468, 4.973649000674243, 5.354619817212085, 6.021451652152212, 6.357027069206774, 6.697327820630451, 7.357789469358524, 7.774785989556767, 8.392856158943386, 8.792690643437789, 9.316626042833336, 9.732814905684977, 10.05313537763523, 10.25031746221232, 11.08285859593699, 11.38732647745938, 11.95400238625713, 12.29336530498892, 12.89417356637478
