| L(s) = 1 | − 0.473·2-s − 1.77·4-s − 2.56·7-s + 1.78·8-s − 6.16·11-s − 2.13·13-s + 1.21·14-s + 2.70·16-s − 3.16·17-s + 0.356·19-s + 2.91·22-s + 4.21·23-s + 1.00·26-s + 4.55·28-s + 1.68·29-s − 8.25·31-s − 4.85·32-s + 1.49·34-s − 3.63·37-s − 0.168·38-s − 2.73·41-s + 7.67·43-s + 10.9·44-s − 1.99·46-s + 11.4·47-s − 0.430·49-s + 3.78·52-s + ⋯ |
| L(s) = 1 | − 0.334·2-s − 0.888·4-s − 0.968·7-s + 0.631·8-s − 1.85·11-s − 0.591·13-s + 0.324·14-s + 0.676·16-s − 0.768·17-s + 0.0817·19-s + 0.622·22-s + 0.878·23-s + 0.197·26-s + 0.860·28-s + 0.313·29-s − 1.48·31-s − 0.858·32-s + 0.257·34-s − 0.597·37-s − 0.0273·38-s − 0.426·41-s + 1.17·43-s + 1.65·44-s − 0.293·46-s + 1.66·47-s − 0.0615·49-s + 0.525·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5070574112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5070574112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.473T + 2T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 13 | \( 1 + 2.13T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.356T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 + 8.25T + 31T^{2} \) |
| 37 | \( 1 + 3.63T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 9.43T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.0109T + 61T^{2} \) |
| 67 | \( 1 + 0.982T + 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 7.20T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.041087525813320272085167390099, −8.603524623125534614084014438468, −7.50524434152150891045652005375, −7.11908500170678217558939013454, −5.75948377190534970946833370950, −5.18940974835223129927983744077, −4.29671585207682245253933946503, −3.22732172636275883827214305887, −2.30332829111001375439013057095, −0.47398112452799819696800086071,
0.47398112452799819696800086071, 2.30332829111001375439013057095, 3.22732172636275883827214305887, 4.29671585207682245253933946503, 5.18940974835223129927983744077, 5.75948377190534970946833370950, 7.11908500170678217558939013454, 7.50524434152150891045652005375, 8.603524623125534614084014438468, 9.041087525813320272085167390099
