| L(s) = 1 | − 0.765·2-s − 3-s − 0.414·4-s + 1.84·5-s + 0.765·6-s + 1.08·8-s + 9-s − 1.41·10-s + 0.765·11-s + 0.414·12-s + 13-s − 1.84·15-s − 0.414·16-s − 0.765·18-s − 0.765·20-s − 0.585·22-s − 1.08·24-s + 2.41·25-s − 0.765·26-s − 27-s + 1.41·30-s − 0.765·32-s − 0.765·33-s − 0.414·36-s − 39-s + 2·40-s − 1.84·41-s + ⋯ |
| L(s) = 1 | − 0.765·2-s − 3-s − 0.414·4-s + 1.84·5-s + 0.765·6-s + 1.08·8-s + 9-s − 1.41·10-s + 0.765·11-s + 0.414·12-s + 13-s − 1.84·15-s − 0.414·16-s − 0.765·18-s − 0.765·20-s − 0.585·22-s − 1.08·24-s + 2.41·25-s − 0.765·26-s − 27-s + 1.41·30-s − 0.765·32-s − 0.765·33-s − 0.414·36-s − 39-s + 2·40-s − 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7822321962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7822321962\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.765T + T^{2} \) |
| 5 | \( 1 - 1.84T + T^{2} \) |
| 11 | \( 1 - 0.765T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 1.84T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 0.765T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.84T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 - 0.765T + T^{2} \) |
| 89 | \( 1 - 0.765T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.484390672417677364621472333185, −8.916617922411679816531492796314, −8.001041034175138742540413876983, −6.66479965743939924028191899809, −6.41558828007815608167045837928, −5.39934382770780728411258987747, −4.84576792266797974083806352232, −3.64025489381050300478321896470, −1.85980951674698477850504945060, −1.20059707091132428755044961250,
1.20059707091132428755044961250, 1.85980951674698477850504945060, 3.64025489381050300478321896470, 4.84576792266797974083806352232, 5.39934382770780728411258987747, 6.41558828007815608167045837928, 6.66479965743939924028191899809, 8.001041034175138742540413876983, 8.916617922411679816531492796314, 9.484390672417677364621472333185
