| L(s) = 1 | + 1.10·5-s + 1.32·7-s − 11-s − 13-s + 3.88·17-s + 8.25·19-s + 8.18·23-s − 3.78·25-s − 4.09·29-s − 9.46·31-s + 1.46·35-s + 9.58·37-s + 5.36·41-s + 2.35·43-s − 10.2·47-s − 5.24·49-s + 10.9·53-s − 1.10·55-s − 1.12·59-s − 7.78·61-s − 1.10·65-s − 5.02·67-s + 13.3·71-s + 7.73·73-s − 1.32·77-s + 3.04·79-s − 16.5·83-s + ⋯ |
| L(s) = 1 | + 0.493·5-s + 0.500·7-s − 0.301·11-s − 0.277·13-s + 0.942·17-s + 1.89·19-s + 1.70·23-s − 0.756·25-s − 0.761·29-s − 1.69·31-s + 0.246·35-s + 1.57·37-s + 0.837·41-s + 0.359·43-s − 1.49·47-s − 0.749·49-s + 1.50·53-s − 0.148·55-s − 0.146·59-s − 0.996·61-s − 0.136·65-s − 0.613·67-s + 1.58·71-s + 0.905·73-s − 0.150·77-s + 0.342·79-s − 1.81·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.495957999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.495957999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 - 8.25T + 19T^{2} \) |
| 23 | \( 1 - 8.18T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 - 9.58T + 37T^{2} \) |
| 41 | \( 1 - 5.36T + 41T^{2} \) |
| 43 | \( 1 - 2.35T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 7.78T + 61T^{2} \) |
| 67 | \( 1 + 5.02T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 - 3.04T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 4.27T + 97T^{2} \) |
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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
−7.971229301281704452781219988128, −7.59399174604513846837435822473, −6.94382661221208998308363311583, −5.76656627825620524308791336662, −5.43359548402243749306040659468, −4.71297425458119893882557751278, −3.58169350770703548749311554505, −2.89229520817794412584853196632, −1.82680719864694050371620501292, −0.902817563739072317481379056596,
0.902817563739072317481379056596, 1.82680719864694050371620501292, 2.89229520817794412584853196632, 3.58169350770703548749311554505, 4.71297425458119893882557751278, 5.43359548402243749306040659468, 5.76656627825620524308791336662, 6.94382661221208998308363311583, 7.59399174604513846837435822473, 7.971229301281704452781219988128
