| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 2·17-s − 18-s − 19-s + 20-s + 21-s + 4·22-s + 24-s + 25-s + 26-s − 27-s − 28-s + 10·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.221973230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221973230\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.50574092332470, −13.96840812235069, −13.32266563065799, −12.86587087230096, −12.47693607386465, −11.71546896878432, −11.50203356505700, −10.55458989703777, −10.40250095353669, −9.865515146653148, −9.527470363845311, −8.674983112463281, −8.175251456156253, −7.740425745780187, −7.053088325052666, −6.498668550783741, −5.996255148756861, −5.530694209129765, −4.700467071527245, −4.394079008296407, −3.103648940322852, −2.808030260538127, −2.085238726333381, −1.112630158498510, −0.5107329057695889,
0.5107329057695889, 1.112630158498510, 2.085238726333381, 2.808030260538127, 3.103648940322852, 4.394079008296407, 4.700467071527245, 5.530694209129765, 5.996255148756861, 6.498668550783741, 7.053088325052666, 7.740425745780187, 8.175251456156253, 8.674983112463281, 9.527470363845311, 9.865515146653148, 10.40250095353669, 10.55458989703777, 11.50203356505700, 11.71546896878432, 12.47693607386465, 12.86587087230096, 13.32266563065799, 13.96840812235069, 14.50574092332470
