| L(s) = 1 | + 4-s + 5-s − 7-s + 11-s + 13-s + 16-s − 17-s + 20-s + 25-s − 28-s − 2·29-s − 35-s + 44-s − 47-s + 49-s + 52-s + 55-s + 64-s + 65-s − 68-s + 71-s + 73-s − 77-s − 79-s + 80-s − 83-s − 85-s + ⋯ |
| L(s) = 1 | + 4-s + 5-s − 7-s + 11-s + 13-s + 16-s − 17-s + 20-s + 25-s − 28-s − 2·29-s − 35-s + 44-s − 47-s + 49-s + 52-s + 55-s + 64-s + 65-s − 68-s + 71-s + 73-s − 77-s − 79-s + 80-s − 83-s − 85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.804441478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804441478\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 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 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.194700633847233223450224927535, −8.300442887078491936958185406262, −7.12644189093544422929875110759, −6.60559797290224989473112797932, −6.11026485621058705206860222320, −5.43673804937837669674580474972, −4.01630125605223224290586171109, −3.27024455837105961588727135828, −2.25283405348471585089568643309, −1.41645267273269926792731925477,
1.41645267273269926792731925477, 2.25283405348471585089568643309, 3.27024455837105961588727135828, 4.01630125605223224290586171109, 5.43673804937837669674580474972, 6.11026485621058705206860222320, 6.60559797290224989473112797932, 7.12644189093544422929875110759, 8.300442887078491936958185406262, 9.194700633847233223450224927535
