| L(s) = 1 | − 2-s + 3-s + 4-s − 0.695·5-s − 6-s + 7-s − 8-s + 9-s + 0.695·10-s − 2.58·11-s + 12-s − 0.341·13-s − 14-s − 0.695·15-s + 16-s − 6.49·17-s − 18-s + 4.09·19-s − 0.695·20-s + 21-s + 2.58·22-s − 1.17·23-s − 24-s − 4.51·25-s + 0.341·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.311·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.219·10-s − 0.780·11-s + 0.288·12-s − 0.0947·13-s − 0.267·14-s − 0.179·15-s + 0.250·16-s − 1.57·17-s − 0.235·18-s + 0.938·19-s − 0.155·20-s + 0.218·21-s + 0.551·22-s − 0.245·23-s − 0.204·24-s − 0.903·25-s + 0.0669·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.489138579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.489138579\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 + 0.695T + 5T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 0.341T + 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 - 9.17T + 29T^{2} \) |
| 31 | \( 1 - 8.97T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 - 1.40T + 41T^{2} \) |
| 43 | \( 1 - 5.08T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 1.31T + 53T^{2} \) |
| 59 | \( 1 + 6.12T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 + 2.56T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 + 4.67T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.88329757877048018648759392782, −7.49036020599585989900370003814, −6.53656014166988550718222560300, −6.01749619723164297936455121657, −4.73823596880631733640015227768, −4.47799197618134670882051253096, −3.20597910620224286455106251455, −2.62412340660594217138960960185, −1.79201819630368385903622095856, −0.64465874352040433799083522233,
0.64465874352040433799083522233, 1.79201819630368385903622095856, 2.62412340660594217138960960185, 3.20597910620224286455106251455, 4.47799197618134670882051253096, 4.73823596880631733640015227768, 6.01749619723164297936455121657, 6.53656014166988550718222560300, 7.49036020599585989900370003814, 7.88329757877048018648759392782
