| L(s) = 1 | − 5-s − 2.07·7-s − 5.07·11-s − 3.48·13-s − 6.48·17-s − 7.48·19-s − 23-s + 25-s + 1.56·29-s + 0.0791·31-s + 2.07·35-s − 9.71·37-s + 0.480·41-s + 8·43-s − 6.96·47-s − 2.67·49-s − 11.7·53-s + 5.07·55-s + 11.5·59-s + 7.88·61-s + 3.48·65-s − 9.71·67-s + 9.67·71-s − 13.2·73-s + 10.5·77-s − 12.3·79-s + 4.59·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.785·7-s − 1.53·11-s − 0.965·13-s − 1.57·17-s − 1.71·19-s − 0.208·23-s + 0.200·25-s + 0.289·29-s + 0.0142·31-s + 0.351·35-s − 1.59·37-s + 0.0751·41-s + 1.21·43-s − 1.01·47-s − 0.382·49-s − 1.60·53-s + 0.684·55-s + 1.50·59-s + 1.00·61-s + 0.431·65-s − 1.18·67-s + 1.14·71-s − 1.55·73-s + 1.20·77-s − 1.38·79-s + 0.504·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1083942512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1083942512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 + 5.07T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 0.0791T + 31T^{2} \) |
| 37 | \( 1 + 9.71T + 37T^{2} \) |
| 41 | \( 1 - 0.480T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 6.96T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 + 9.71T + 67T^{2} \) |
| 71 | \( 1 - 9.67T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 4.59T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 - 7.23T + 97T^{2} \) |
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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
−7.79634044404134290654461363384, −7.05863636150598934177994427581, −6.55913264205417122304155305226, −5.77672117297107931255997652604, −4.83000807713066451680847828344, −4.43618959967067953824648731457, −3.43368101085530767956384071677, −2.58546679564020879664825248617, −2.04832530416573198215262039364, −0.14697329336062861772474524143,
0.14697329336062861772474524143, 2.04832530416573198215262039364, 2.58546679564020879664825248617, 3.43368101085530767956384071677, 4.43618959967067953824648731457, 4.83000807713066451680847828344, 5.77672117297107931255997652604, 6.55913264205417122304155305226, 7.05863636150598934177994427581, 7.79634044404134290654461363384
