| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s − 13-s − 15-s + 17-s − 21-s + 23-s + 25-s − 27-s + 33-s + 35-s + 37-s + 39-s − 41-s + 45-s − 51-s + 53-s − 55-s + 2·59-s − 61-s + 63-s − 65-s − 2·67-s − 69-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s − 13-s − 15-s + 17-s − 21-s + 23-s + 25-s − 27-s + 33-s + 35-s + 37-s + 39-s − 41-s + 45-s − 51-s + 53-s − 55-s + 2·59-s − 61-s + 63-s − 65-s − 2·67-s − 69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9012196851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9012196851\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 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 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−10.30972096544066562026158851175, −10.08166971564972611122635486197, −8.907747183540318117793516215202, −7.72981625574447508713641658946, −7.06390782928906118278810892802, −5.83731050284631645753863034111, −5.23839022199601116604574165822, −4.59123019940096217763618718273, −2.73186360104816505567535766006, −1.43808989056054881699993264487,
1.43808989056054881699993264487, 2.73186360104816505567535766006, 4.59123019940096217763618718273, 5.23839022199601116604574165822, 5.83731050284631645753863034111, 7.06390782928906118278810892802, 7.72981625574447508713641658946, 8.907747183540318117793516215202, 10.08166971564972611122635486197, 10.30972096544066562026158851175
