| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s + 14-s + 16-s + 2·17-s − 19-s − 20-s − 22-s − 23-s + 28-s − 31-s + 32-s + 2·34-s − 35-s − 37-s − 38-s − 40-s − 41-s − 44-s − 46-s + 49-s + 55-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s + 14-s + 16-s + 2·17-s − 19-s − 20-s − 22-s − 23-s + 28-s − 31-s + 32-s + 2·34-s − 35-s − 37-s − 38-s − 40-s − 41-s − 44-s − 46-s + 49-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.607653798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.607653798\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−10.70307280364907865682803440503, −10.12581627018661653478468810629, −8.396217773786489877770017604571, −7.83589544027696710782383289890, −7.22899359841526502853579716849, −5.82245334318382534580430459072, −5.12071423778698502896077546498, −4.14180636693353008858917816235, −3.27624025963414795564603285857, −1.86688192482841846767665207105,
1.86688192482841846767665207105, 3.27624025963414795564603285857, 4.14180636693353008858917816235, 5.12071423778698502896077546498, 5.82245334318382534580430459072, 7.22899359841526502853579716849, 7.83589544027696710782383289890, 8.396217773786489877770017604571, 10.12581627018661653478468810629, 10.70307280364907865682803440503
