| L(s) = 1 | − 2.43·2-s + 3.94·4-s − 5-s + 3.59·7-s − 4.75·8-s + 2.43·10-s + 6.58·11-s + 2.25·13-s − 8.77·14-s + 3.69·16-s + 0.256·17-s + 7.95·19-s − 3.94·20-s − 16.0·22-s + 0.961·23-s + 25-s − 5.50·26-s + 14.2·28-s + 29-s − 3.95·31-s + 0.498·32-s − 0.624·34-s − 3.59·35-s − 3.76·37-s − 19.3·38-s + 4.75·40-s + 5.00·41-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.97·4-s − 0.447·5-s + 1.35·7-s − 1.67·8-s + 0.771·10-s + 1.98·11-s + 0.625·13-s − 2.34·14-s + 0.922·16-s + 0.0621·17-s + 1.82·19-s − 0.882·20-s − 3.42·22-s + 0.200·23-s + 0.200·25-s − 1.07·26-s + 2.68·28-s + 0.185·29-s − 0.709·31-s + 0.0881·32-s − 0.107·34-s − 0.607·35-s − 0.619·37-s − 3.14·38-s + 0.751·40-s + 0.781·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9932835957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9932835957\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 - 6.58T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 - 0.256T + 17T^{2} \) |
| 19 | \( 1 - 7.95T + 19T^{2} \) |
| 23 | \( 1 - 0.961T + 23T^{2} \) |
| 31 | \( 1 + 3.95T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 - 5.00T + 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 + 7.02T + 47T^{2} \) |
| 53 | \( 1 - 6.38T + 53T^{2} \) |
| 59 | \( 1 + 3.19T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 5.22T + 79T^{2} \) |
| 83 | \( 1 - 0.195T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 + 2.97T + 97T^{2} \) |
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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
−9.419145739021430289570199526057, −8.820006957905320064638999668462, −8.243302321430182538589093562683, −7.38353278976939301962074410580, −6.86395505317224896384863573521, −5.70606284177804521486766662382, −4.44372285388301507122802994511, −3.32115702740899757113288754156, −1.65259694094452858371360365222, −1.09860485634575322802576802578,
1.09860485634575322802576802578, 1.65259694094452858371360365222, 3.32115702740899757113288754156, 4.44372285388301507122802994511, 5.70606284177804521486766662382, 6.86395505317224896384863573521, 7.38353278976939301962074410580, 8.243302321430182538589093562683, 8.820006957905320064638999668462, 9.419145739021430289570199526057
