| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s − 11-s + 13-s + 16-s + 18-s + 19-s − 20-s − 22-s − 23-s + 25-s + 26-s + 32-s + 36-s − 37-s + 38-s − 40-s + 41-s − 44-s − 45-s − 46-s + 47-s + 50-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s − 11-s + 13-s + 16-s + 18-s + 19-s − 20-s − 22-s − 23-s + 25-s + 26-s + 32-s + 36-s − 37-s + 38-s − 40-s + 41-s − 44-s − 45-s − 46-s + 47-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.022700445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022700445\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.458442909184644190448952985810, −8.227540009699451793622756686290, −7.66995394950764974660213616749, −7.05766719720661597993797872211, −6.10090393195841991817926275197, −5.22245196487187916817760030421, −4.35970079785711929259445751653, −3.71055737062696087996366167137, −2.82378041608851767887299041034, −1.43310969710612178506045447039,
1.43310969710612178506045447039, 2.82378041608851767887299041034, 3.71055737062696087996366167137, 4.35970079785711929259445751653, 5.22245196487187916817760030421, 6.10090393195841991817926275197, 7.05766719720661597993797872211, 7.66995394950764974660213616749, 8.227540009699451793622756686290, 9.458442909184644190448952985810
