| L(s) = 1 | − 2.30·2-s + 3.30·4-s + 4.30·7-s − 3.00·8-s − 5.30·11-s − 0.302·13-s − 9.90·14-s + 0.302·16-s − 2.30·17-s − 7.21·19-s + 12.2·22-s + 6.90·23-s + 0.697·26-s + 14.2·28-s + 1.39·29-s + 8.21·31-s + 5.30·32-s + 5.30·34-s − 1.90·37-s + 16.6·38-s + 5.30·41-s + 5.69·43-s − 17.5·44-s − 15.9·46-s − 6.21·47-s + 11.5·49-s − 1.00·52-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.65·4-s + 1.62·7-s − 1.06·8-s − 1.59·11-s − 0.0839·13-s − 2.64·14-s + 0.0756·16-s − 0.558·17-s − 1.65·19-s + 2.60·22-s + 1.44·23-s + 0.136·26-s + 2.68·28-s + 0.258·29-s + 1.47·31-s + 0.937·32-s + 0.909·34-s − 0.313·37-s + 2.69·38-s + 0.828·41-s + 0.868·43-s − 2.64·44-s − 2.34·46-s − 0.905·47-s + 1.64·49-s − 0.138·52-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
| \(L(1)\) |
\(\approx\) |
\(0.7841636429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7841636429\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.30T + 2T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 + 0.302T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 - 8.21T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 - 5.30T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 - 3.69T + 59T^{2} \) |
| 61 | \( 1 + 0.302T + 61T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 - 9.60T + 73T^{2} \) |
| 79 | \( 1 + 2.60T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.809206647398376497595043170181, −8.480453466489628367403068311486, −7.84134289604426828656476669863, −7.23474130489197555558553914900, −6.26742063818253214884215499158, −5.03931400311623492449024173076, −4.49282917038491840109831615849, −2.62915765805895904633698583374, −2.00647929830804394027415360413, −0.74851113578160851277905693506,
0.74851113578160851277905693506, 2.00647929830804394027415360413, 2.62915765805895904633698583374, 4.49282917038491840109831615849, 5.03931400311623492449024173076, 6.26742063818253214884215499158, 7.23474130489197555558553914900, 7.84134289604426828656476669863, 8.480453466489628367403068311486, 8.809206647398376497595043170181
