| L(s) = 1 | + 2-s + 0.347·3-s + 4-s + 0.347·6-s + 1.87·7-s + 8-s − 0.879·9-s + 0.347·12-s + 1.53·13-s + 1.87·14-s + 16-s − 1.53·17-s − 0.879·18-s − 19-s + 0.652·21-s − 0.347·23-s + 0.347·24-s + 1.53·26-s − 0.652·27-s + 1.87·28-s − 1.53·29-s + 32-s − 1.53·34-s − 0.879·36-s − 37-s − 38-s + 0.532·39-s + ⋯ |
| L(s) = 1 | + 2-s + 0.347·3-s + 4-s + 0.347·6-s + 1.87·7-s + 8-s − 0.879·9-s + 0.347·12-s + 1.53·13-s + 1.87·14-s + 16-s − 1.53·17-s − 0.879·18-s − 19-s + 0.652·21-s − 0.347·23-s + 0.347·24-s + 1.53·26-s − 0.652·27-s + 1.87·28-s − 1.53·29-s + 32-s − 1.53·34-s − 0.879·36-s − 37-s − 38-s + 0.532·39-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.194123755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.194123755\) |
| \(L(1)\) |
|
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 | \( 1 - 0.347T + T^{2} \) |
| 7 | \( 1 - 1.87T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.53T + T^{2} \) |
| 17 | \( 1 + 1.53T + T^{2} \) |
| 23 | \( 1 + 0.347T + T^{2} \) |
| 29 | \( 1 + 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + 1.87T + T^{2} \) |
| 59 | \( 1 + 0.347T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.347T + T^{2} \) |
| 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
−8.590342833621538443693486230975, −7.963440350336301420218463122734, −7.21416955681902813711862209166, −6.16550787717358948287411763094, −5.71834719527369979461737005595, −4.74236798352515693945149310771, −4.20753745430707873500565046856, −3.35691748167187855035336346157, −2.14536010350015456596867838738, −1.68694758213333139295167510214,
1.68694758213333139295167510214, 2.14536010350015456596867838738, 3.35691748167187855035336346157, 4.20753745430707873500565046856, 4.74236798352515693945149310771, 5.71834719527369979461737005595, 6.16550787717358948287411763094, 7.21416955681902813711862209166, 7.963440350336301420218463122734, 8.590342833621538443693486230975
