| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8141483777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8141483777\) |
| \(L(1)\) |
|
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
−10.75010751277532928625464750946, −9.396523660145784083953375779468, −9.021317239565700150672173967860, −8.240497659360209369066790518809, −7.00908279562381707407311891836, −6.44832622157380447205612607315, −5.38469650386639956657124971674, −4.13301583144023729075370646114, −2.37186577443185643307235302846, −1.60535073567869002038302573940,
1.60535073567869002038302573940, 2.37186577443185643307235302846, 4.13301583144023729075370646114, 5.38469650386639956657124971674, 6.44832622157380447205612607315, 7.00908279562381707407311891836, 8.240497659360209369066790518809, 9.021317239565700150672173967860, 9.396523660145784083953375779468, 10.75010751277532928625464750946
