L(s) = 1 | + 1.53·2-s − 1.96·3-s + 1.34·4-s + 1.28·5-s − 3.01·6-s + 7-s + 0.532·8-s + 2.87·9-s + 1.96·10-s − 11-s − 2.65·12-s + 1.73·13-s + 1.53·14-s − 2.53·15-s − 0.532·16-s − 0.684·17-s + 4.41·18-s + 1.73·20-s − 1.96·21-s − 1.53·22-s + 23-s − 1.04·24-s + 0.652·25-s + 2.65·26-s − 3.70·27-s + 1.34·28-s − 0.347·29-s + ⋯ |
L(s) = 1 | + 1.53·2-s − 1.96·3-s + 1.34·4-s + 1.28·5-s − 3.01·6-s + 7-s + 0.532·8-s + 2.87·9-s + 1.96·10-s − 11-s − 2.65·12-s + 1.73·13-s + 1.53·14-s − 2.53·15-s − 0.532·16-s − 0.684·17-s + 4.41·18-s + 1.73·20-s − 1.96·21-s − 1.53·22-s + 23-s − 1.04·24-s + 0.652·25-s + 2.65·26-s − 3.70·27-s + 1.34·28-s − 0.347·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.209020093\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.209020093\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 - 1.53T + T^{2} \) |
| 3 | \( 1 + 1.96T + T^{2} \) |
| 5 | \( 1 - 1.28T + T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 + 0.684T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + 0.347T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.684T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.87T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.53T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.96T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−8.898928529603094074131830040419, −7.68804865044099508765296159335, −6.63692396953606996417809767637, −6.25759219855834796251004827812, −5.60712571370245991408071246588, −5.11295444155801005289922682034, −4.65302670366272961243435162874, −3.66365814683845920905785382330, −2.19715821318105808118061550971, −1.31928300792488560915036110146,
1.31928300792488560915036110146, 2.19715821318105808118061550971, 3.66365814683845920905785382330, 4.65302670366272961243435162874, 5.11295444155801005289922682034, 5.60712571370245991408071246588, 6.25759219855834796251004827812, 6.63692396953606996417809767637, 7.68804865044099508765296159335, 8.898928529603094074131830040419