L(s) = 1 | − 0.397·3-s + 3.51·5-s + 2.09·7-s − 2.84·9-s + 2.13·11-s + 5.28·13-s − 1.39·15-s − 17-s − 0.313·19-s − 0.832·21-s + 1.96·23-s + 7.36·25-s + 2.32·27-s − 3.78·29-s + 1.00·31-s − 0.847·33-s + 7.35·35-s − 1.71·37-s − 2.10·39-s + 8.45·41-s + 6.98·43-s − 9.99·45-s − 2.81·47-s − 2.62·49-s + 0.397·51-s − 1.10·53-s + 7.49·55-s + ⋯ |
L(s) = 1 | − 0.229·3-s + 1.57·5-s + 0.790·7-s − 0.947·9-s + 0.642·11-s + 1.46·13-s − 0.361·15-s − 0.242·17-s − 0.0718·19-s − 0.181·21-s + 0.409·23-s + 1.47·25-s + 0.447·27-s − 0.701·29-s + 0.180·31-s − 0.147·33-s + 1.24·35-s − 0.282·37-s − 0.336·39-s + 1.32·41-s + 1.06·43-s − 1.48·45-s − 0.410·47-s − 0.374·49-s + 0.0556·51-s − 0.151·53-s + 1.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.866040869\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.866040869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 0.397T + 3T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 - 2.09T + 7T^{2} \) |
| 11 | \( 1 - 2.13T + 11T^{2} \) |
| 13 | \( 1 - 5.28T + 13T^{2} \) |
| 19 | \( 1 + 0.313T + 19T^{2} \) |
| 23 | \( 1 - 1.96T + 23T^{2} \) |
| 29 | \( 1 + 3.78T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 + 1.71T + 37T^{2} \) |
| 41 | \( 1 - 8.45T + 41T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 + 2.81T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 1.96T + 67T^{2} \) |
| 71 | \( 1 - 5.84T + 71T^{2} \) |
| 73 | \( 1 + 9.52T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 6.74T + 89T^{2} \) |
| 97 | \( 1 + 0.816T + 97T^{2} \) |
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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
−8.683140000199809132055712659982, −7.81438006841588965391461553969, −6.71132216897011695767151604336, −6.06898072442235456619875928445, −5.67502717293486038531118895775, −4.87875080687725096768085307036, −3.86534484912854790658456248384, −2.79120308628969434580199445230, −1.86583601681734882667885909121, −1.07225959158841133850026181839,
1.07225959158841133850026181839, 1.86583601681734882667885909121, 2.79120308628969434580199445230, 3.86534484912854790658456248384, 4.87875080687725096768085307036, 5.67502717293486038531118895775, 6.06898072442235456619875928445, 6.71132216897011695767151604336, 7.81438006841588965391461553969, 8.683140000199809132055712659982