| L(s) = 1 | − 2-s + 4-s − 2·5-s − 7-s − 8-s + 2·10-s − 4·11-s − 2·13-s + 14-s + 16-s − 6·17-s − 4·19-s − 2·20-s + 4·22-s − 25-s + 2·26-s − 28-s + 2·29-s − 8·31-s − 32-s + 6·34-s + 2·35-s − 6·37-s + 4·38-s + 2·40-s + 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.338·35-s − 0.986·37-s + 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.60765318405394, −13.92739861877108, −13.35883147110692, −12.65541876065798, −12.61236484434942, −11.87462251280063, −11.21359663719638, −10.94022833424217, −10.40187780162971, −9.986669649430456, −9.160888520806541, −8.865279047797468, −8.302441560872297, −7.756519312004137, −7.240518245404569, −6.923693727975659, −6.171919609289948, −5.544374190923171, −4.950714699983597, −4.104063589260545, −3.873801712240748, −2.784562876575408, −2.462051316141086, −1.761328912879763, −0.5164299035041065, 0,
0.5164299035041065, 1.761328912879763, 2.462051316141086, 2.784562876575408, 3.873801712240748, 4.104063589260545, 4.950714699983597, 5.544374190923171, 6.171919609289948, 6.923693727975659, 7.240518245404569, 7.756519312004137, 8.302441560872297, 8.865279047797468, 9.160888520806541, 9.986669649430456, 10.40187780162971, 10.94022833424217, 11.21359663719638, 11.87462251280063, 12.61236484434942, 12.65541876065798, 13.35883147110692, 13.92739861877108, 14.60765318405394
