| L(s) = 1 | + 0.225·2-s − 7.94·4-s + 5·5-s − 31.1·7-s − 3.59·8-s + 1.12·10-s − 18.1·11-s − 50.1·13-s − 7.02·14-s + 62.7·16-s + 131.·17-s + 23.2·19-s − 39.7·20-s − 4.08·22-s + 32.9·23-s + 25·25-s − 11.2·26-s + 247.·28-s − 125.·29-s + 125.·31-s + 42.8·32-s + 29.6·34-s − 155.·35-s + 99.9·37-s + 5.23·38-s − 17.9·40-s − 245.·41-s + ⋯ |
| L(s) = 1 | + 0.0796·2-s − 0.993·4-s + 0.447·5-s − 1.68·7-s − 0.158·8-s + 0.0356·10-s − 0.496·11-s − 1.07·13-s − 0.134·14-s + 0.981·16-s + 1.87·17-s + 0.280·19-s − 0.444·20-s − 0.0395·22-s + 0.299·23-s + 0.200·25-s − 0.0852·26-s + 1.67·28-s − 0.805·29-s + 0.724·31-s + 0.236·32-s + 0.149·34-s − 0.753·35-s + 0.444·37-s + 0.0223·38-s − 0.0710·40-s − 0.934·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.058538194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058538194\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 0.225T + 8T^{2} \) |
| 7 | \( 1 + 31.1T + 343T^{2} \) |
| 11 | \( 1 + 18.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 32.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 99.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 245.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 139.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 472.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 421.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 742.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 8.97T + 2.26e5T^{2} \) |
| 67 | \( 1 - 588.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 48.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 409.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 530.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 294.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 852.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 388.T + 9.12e5T^{2} \) |
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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
−10.41848777330745995747250579784, −9.718574894443736854808226081763, −9.426404286835289086574115529521, −8.093104052849926694559316666476, −7.08162077865792301350826944255, −5.84955259337724893680988145165, −5.15546169278778745004258186009, −3.70795399960364442790559628130, −2.79306455443702338177852482022, −0.65008926217116654458751986923,
0.65008926217116654458751986923, 2.79306455443702338177852482022, 3.70795399960364442790559628130, 5.15546169278778745004258186009, 5.84955259337724893680988145165, 7.08162077865792301350826944255, 8.093104052849926694559316666476, 9.426404286835289086574115529521, 9.718574894443736854808226081763, 10.41848777330745995747250579784
