| L(s) = 1 | − 2-s + 4-s − 0.305·5-s − 3.02·7-s − 8-s + 0.305·10-s − 2.44·11-s − 2.87·13-s + 3.02·14-s + 16-s − 7.10·17-s − 2.23·19-s − 0.305·20-s + 2.44·22-s − 4.90·25-s + 2.87·26-s − 3.02·28-s + 3.30·29-s − 3.46·31-s − 32-s + 7.10·34-s + 0.924·35-s + 3.02·37-s + 2.23·38-s + 0.305·40-s − 4.05·41-s − 6.30·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.136·5-s − 1.14·7-s − 0.353·8-s + 0.0965·10-s − 0.738·11-s − 0.797·13-s + 0.809·14-s + 0.250·16-s − 1.72·17-s − 0.513·19-s − 0.0682·20-s + 0.522·22-s − 0.981·25-s + 0.563·26-s − 0.572·28-s + 0.613·29-s − 0.622·31-s − 0.176·32-s + 1.21·34-s + 0.156·35-s + 0.498·37-s + 0.362·38-s + 0.0482·40-s − 0.633·41-s − 0.961·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09405609149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09405609149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 0.305T + 5T^{2} \) |
| 7 | \( 1 + 3.02T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 + 7.10T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 - 3.89T + 47T^{2} \) |
| 53 | \( 1 + 6.16T + 53T^{2} \) |
| 59 | \( 1 - 0.622T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 2.61T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 2.07T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 + 9.57T + 89T^{2} \) |
| 97 | \( 1 + 9.87T + 97T^{2} \) |
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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
−7.61287018022069390773907223592, −7.12032564273242542797021702057, −6.41414032433393486517697411593, −5.92759527140195567462722832802, −4.89287967151509087625953157703, −4.20919286169587285932313348064, −3.19340281440398687267888031744, −2.55642933216099468438077311874, −1.77934002057206242163037134062, −0.15161895261114972287139250364,
0.15161895261114972287139250364, 1.77934002057206242163037134062, 2.55642933216099468438077311874, 3.19340281440398687267888031744, 4.20919286169587285932313348064, 4.89287967151509087625953157703, 5.92759527140195567462722832802, 6.41414032433393486517697411593, 7.12032564273242542797021702057, 7.61287018022069390773907223592
