| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 6·11-s − 12-s − 13-s − 15-s + 16-s − 18-s − 8·19-s + 20-s − 6·22-s + 24-s + 25-s + 26-s − 27-s − 6·29-s + 30-s − 2·31-s − 32-s − 6·33-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.223·20-s − 1.27·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.11·29-s + 0.182·30-s − 0.359·31-s − 0.176·32-s − 1.04·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−16.51968672814725, −15.25761740686544, −15.04203216106226, −14.57415535979741, −13.77347995838872, −13.24395555687491, −12.43898483989886, −12.15799571917868, −11.45264814274749, −11.00928323967556, −10.39451614855780, −9.893914301811817, −9.113882205546813, −8.949912229438682, −8.192390535161066, −7.301698385191597, −6.833601068859039, −6.275509880832092, −5.833903046939745, −4.984512822184941, −4.101757405047359, −3.703794212981875, −2.444577458373905, −1.819556041604228, −1.073408260765778, 0,
1.073408260765778, 1.819556041604228, 2.444577458373905, 3.703794212981875, 4.101757405047359, 4.984512822184941, 5.833903046939745, 6.275509880832092, 6.833601068859039, 7.301698385191597, 8.192390535161066, 8.949912229438682, 9.113882205546813, 9.893914301811817, 10.39451614855780, 11.00928323967556, 11.45264814274749, 12.15799571917868, 12.43898483989886, 13.24395555687491, 13.77347995838872, 14.57415535979741, 15.04203216106226, 15.25761740686544, 16.51968672814725
