| L(s) = 1 | − 2.80·2-s + 0.208·3-s + 5.84·4-s − 0.584·6-s + 7-s − 10.7·8-s − 2.95·9-s − 0.0871·11-s + 1.21·12-s − 6.31·13-s − 2.80·14-s + 18.4·16-s + 1.93·17-s + 8.27·18-s + 6.29·19-s + 0.208·21-s + 0.243·22-s + 23-s − 2.24·24-s + 17.6·26-s − 1.24·27-s + 5.84·28-s + 1.14·29-s + 0.488·31-s − 30.1·32-s − 0.0181·33-s − 5.40·34-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.120·3-s + 2.92·4-s − 0.238·6-s + 0.377·7-s − 3.80·8-s − 0.985·9-s − 0.0262·11-s + 0.351·12-s − 1.75·13-s − 0.748·14-s + 4.60·16-s + 0.468·17-s + 1.95·18-s + 1.44·19-s + 0.0455·21-s + 0.0520·22-s + 0.208·23-s − 0.458·24-s + 3.46·26-s − 0.239·27-s + 1.10·28-s + 0.212·29-s + 0.0877·31-s − 5.32·32-s − 0.00316·33-s − 0.927·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5732828500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5732828500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 3 | \( 1 - 0.208T + 3T^{2} \) |
| 11 | \( 1 + 0.0871T + 11T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 - 0.488T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 - 0.581T + 43T^{2} \) |
| 47 | \( 1 + 7.80T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 + 5.74T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 - 8.56T + 67T^{2} \) |
| 71 | \( 1 - 5.86T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 0.623T + 97T^{2} \) |
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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
−8.487502174645255857472646503448, −7.68486561420080386685372223677, −7.51326314589243582082470226624, −6.56937086012743382995813296769, −5.72075193637219418837107892901, −4.95987858011245365712694950272, −3.18327367376050649969398980104, −2.70137608844489302643727216421, −1.71933606945939416595009483271, −0.56801985540103914822977386212,
0.56801985540103914822977386212, 1.71933606945939416595009483271, 2.70137608844489302643727216421, 3.18327367376050649969398980104, 4.95987858011245365712694950272, 5.72075193637219418837107892901, 6.56937086012743382995813296769, 7.51326314589243582082470226624, 7.68486561420080386685372223677, 8.487502174645255857472646503448
