| L(s) = 1 | + 3-s + 1.43·5-s − 7-s + 9-s + 11-s + 3.95·13-s + 1.43·15-s + 4.51·17-s + 3·19-s − 21-s + 23-s − 2.95·25-s + 27-s − 10.4·29-s + 0.344·31-s + 33-s − 1.43·35-s − 7.55·37-s + 3.95·39-s + 5.17·41-s + 7.08·43-s + 1.43·45-s + 7.51·47-s + 49-s + 4.51·51-s − 4.95·53-s + 1.43·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.640·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s + 1.09·13-s + 0.369·15-s + 1.09·17-s + 0.688·19-s − 0.218·21-s + 0.208·23-s − 0.590·25-s + 0.192·27-s − 1.93·29-s + 0.0618·31-s + 0.174·33-s − 0.241·35-s − 1.24·37-s + 0.632·39-s + 0.808·41-s + 1.08·43-s + 0.213·45-s + 1.09·47-s + 0.142·49-s + 0.632·51-s − 0.679·53-s + 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.610049905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.610049905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 1.43T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 0.344T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 - 7.08T + 43T^{2} \) |
| 47 | \( 1 - 7.51T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 - 4.60T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 0.518T + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 + 6.41T + 83T^{2} \) |
| 89 | \( 1 + 2.81T + 89T^{2} \) |
| 97 | \( 1 + 6.51T + 97T^{2} \) |
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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
−9.407228447608093271629059552644, −8.479606704049935851441019024250, −7.65735134226746258376515127054, −6.90961051210619885861043611609, −5.87246964237036430892775570673, −5.42296819492432019186914047083, −3.93706893687140396564020593533, −3.41655375162077889243052086018, −2.22340547847886957912197201168, −1.15625421902023932629983090111,
1.15625421902023932629983090111, 2.22340547847886957912197201168, 3.41655375162077889243052086018, 3.93706893687140396564020593533, 5.42296819492432019186914047083, 5.87246964237036430892775570673, 6.90961051210619885861043611609, 7.65735134226746258376515127054, 8.479606704049935851441019024250, 9.407228447608093271629059552644
