| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s + 5·11-s + 12-s + 13-s − 2·14-s + 15-s + 16-s − 5·17-s − 18-s + 2·19-s + 20-s + 2·21-s − 5·22-s + 4·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.436·21-s − 1.06·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.778768071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.778768071\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.59850133887873, −15.98891625046216, −15.14006809553578, −14.87782216276507, −14.21226512282698, −13.80432969580846, −13.00429559494537, −12.50987329477036, −11.58298162075043, −11.18575651571984, −10.78972576015295, −9.726163250874489, −9.403868847321580, −8.851337074565675, −8.386866351380008, −7.623312657236658, −6.904905827624039, −6.512938422814020, −5.627425001199713, −4.795715610208073, −4.016293995926330, −3.298301095920374, −2.254653978695235, −1.684455034414420, −0.8841854385906121,
0.8841854385906121, 1.684455034414420, 2.254653978695235, 3.298301095920374, 4.016293995926330, 4.795715610208073, 5.627425001199713, 6.512938422814020, 6.904905827624039, 7.623312657236658, 8.386866351380008, 8.851337074565675, 9.403868847321580, 9.726163250874489, 10.78972576015295, 11.18575651571984, 11.58298162075043, 12.50987329477036, 13.00429559494537, 13.80432969580846, 14.21226512282698, 14.87782216276507, 15.14006809553578, 15.98891625046216, 16.59850133887873
