| L(s) = 1 | + 2-s + 1.53·3-s + 4-s + 1.53·6-s − 0.347·7-s + 8-s + 1.34·9-s + 1.53·12-s − 1.87·13-s − 0.347·14-s + 16-s + 1.87·17-s + 1.34·18-s − 19-s − 0.532·21-s − 1.53·23-s + 1.53·24-s − 1.87·26-s + 0.532·27-s − 0.347·28-s + 1.87·29-s + 32-s + 1.87·34-s + 1.34·36-s − 37-s − 38-s − 2.87·39-s + ⋯ |
| L(s) = 1 | + 2-s + 1.53·3-s + 4-s + 1.53·6-s − 0.347·7-s + 8-s + 1.34·9-s + 1.53·12-s − 1.87·13-s − 0.347·14-s + 16-s + 1.87·17-s + 1.34·18-s − 19-s − 0.532·21-s − 1.53·23-s + 1.53·24-s − 1.87·26-s + 0.532·27-s − 0.347·28-s + 1.87·29-s + 32-s + 1.87·34-s + 1.34·36-s − 37-s − 38-s − 2.87·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.826659113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.826659113\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.53T + T^{2} \) |
| 7 | \( 1 + 0.347T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.87T + T^{2} \) |
| 17 | \( 1 - 1.87T + T^{2} \) |
| 23 | \( 1 + 1.53T + T^{2} \) |
| 29 | \( 1 - 1.87T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 + 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.53T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.430386868846505673501221688348, −7.84766650741892632991739127744, −7.33899857033257092072541227421, −6.50620957546656860262610578358, −5.59338816977122144368062773359, −4.68266859244379119676910764480, −3.97349981518851229136937168876, −3.09016778912144398289337552574, −2.60384243654346506981579542080, −1.72408719718483107940422132223,
1.72408719718483107940422132223, 2.60384243654346506981579542080, 3.09016778912144398289337552574, 3.97349981518851229136937168876, 4.68266859244379119676910764480, 5.59338816977122144368062773359, 6.50620957546656860262610578358, 7.33899857033257092072541227421, 7.84766650741892632991739127744, 8.430386868846505673501221688348
