L(s) = 1 | + 2-s + 1.31·3-s + 4-s + 5-s + 1.31·6-s + 3.61·7-s + 8-s − 1.28·9-s + 10-s + 3.14·11-s + 1.31·12-s + 3.52·13-s + 3.61·14-s + 1.31·15-s + 16-s − 4.96·17-s − 1.28·18-s + 1.16·19-s + 20-s + 4.74·21-s + 3.14·22-s + 1.31·24-s + 25-s + 3.52·26-s − 5.61·27-s + 3.61·28-s + 5.04·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.757·3-s + 0.5·4-s + 0.447·5-s + 0.535·6-s + 1.36·7-s + 0.353·8-s − 0.426·9-s + 0.316·10-s + 0.949·11-s + 0.378·12-s + 0.976·13-s + 0.966·14-s + 0.338·15-s + 0.250·16-s − 1.20·17-s − 0.301·18-s + 0.266·19-s + 0.223·20-s + 1.03·21-s + 0.671·22-s + 0.267·24-s + 0.200·25-s + 0.690·26-s − 1.08·27-s + 0.683·28-s + 0.936·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.690648814\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.690648814\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 - 0.957T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 - 8.77T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 + 7.30T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 1.58T + 89T^{2} \) |
| 97 | \( 1 - 0.135T + 97T^{2} \) |
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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
−8.430861157809439597413043828534, −7.54000185572927038425435840439, −6.50751997536852324732461212976, −6.16257944575604165430639628630, −5.08572635180123879632674136814, −4.54309046642338691765194620559, −3.72212165697646335616116967733, −2.87111354196399107895921625862, −1.99311560665768692724122785565, −1.27390586879148220678242162873,
1.27390586879148220678242162873, 1.99311560665768692724122785565, 2.87111354196399107895921625862, 3.72212165697646335616116967733, 4.54309046642338691765194620559, 5.08572635180123879632674136814, 6.16257944575604165430639628630, 6.50751997536852324732461212976, 7.54000185572927038425435840439, 8.430861157809439597413043828534