| L(s) = 1 | − 0.166·2-s + 3-s − 1.97·4-s + 3.74·5-s − 0.166·6-s − 0.361·7-s + 0.661·8-s + 9-s − 0.624·10-s − 0.314·11-s − 1.97·12-s + 13-s + 0.0601·14-s + 3.74·15-s + 3.83·16-s + 5.97·17-s − 0.166·18-s − 4.10·19-s − 7.39·20-s − 0.361·21-s + 0.0523·22-s + 6.88·23-s + 0.661·24-s + 9.05·25-s − 0.166·26-s + 27-s + 0.712·28-s + ⋯ |
| L(s) = 1 | − 0.117·2-s + 0.577·3-s − 0.986·4-s + 1.67·5-s − 0.0679·6-s − 0.136·7-s + 0.233·8-s + 0.333·9-s − 0.197·10-s − 0.0948·11-s − 0.569·12-s + 0.277·13-s + 0.0160·14-s + 0.967·15-s + 0.958·16-s + 1.44·17-s − 0.0392·18-s − 0.941·19-s − 1.65·20-s − 0.0788·21-s + 0.0111·22-s + 1.43·23-s + 0.134·24-s + 1.81·25-s − 0.0326·26-s + 0.192·27-s + 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.665321714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665321714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.166T + 2T^{2} \) |
| 5 | \( 1 - 3.74T + 5T^{2} \) |
| 7 | \( 1 + 0.361T + 7T^{2} \) |
| 11 | \( 1 + 0.314T + 11T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 + 4.10T + 19T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 + 1.86T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 + 5.66T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 + 7.01T + 61T^{2} \) |
| 67 | \( 1 - 0.855T + 67T^{2} \) |
| 71 | \( 1 - 9.35T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 - 0.529T + 79T^{2} \) |
| 83 | \( 1 + 0.539T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 4.76T + 97T^{2} \) |
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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
−8.651813096334753246286843422874, −7.940339415326440838191561899987, −6.93957005264537953684691368137, −6.17228547346101726414445355347, −5.34656205435694731464563283581, −4.89989226350016278432988260280, −3.71371288133943885482864072103, −2.96339242398092944744735933138, −1.89420301528576508864178230139, −1.00282145841893141178783107374,
1.00282145841893141178783107374, 1.89420301528576508864178230139, 2.96339242398092944744735933138, 3.71371288133943885482864072103, 4.89989226350016278432988260280, 5.34656205435694731464563283581, 6.17228547346101726414445355347, 6.93957005264537953684691368137, 7.940339415326440838191561899987, 8.651813096334753246286843422874
