| L(s) = 1 | − 1.73i·3-s + (−2 − 1.73i)7-s − 2.99·9-s + 6.92i·13-s − 3.46i·19-s + (−2.99 + 3.46i)21-s + 5.19i·27-s + 10.3i·31-s − 10·37-s + 11.9·39-s + 8·43-s + (1.00 + 6.92i)49-s − 5.99·57-s + 6.92i·61-s + (5.99 + 5.19i)63-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + (−0.755 − 0.654i)7-s − 0.999·9-s + 1.92i·13-s − 0.794i·19-s + (−0.654 + 0.755i)21-s + 0.999i·27-s + 1.86i·31-s − 1.64·37-s + 1.92·39-s + 1.21·43-s + (0.142 + 0.989i)49-s − 0.794·57-s + 0.887i·61-s + (0.755 + 0.654i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8787915032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8787915032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6.92iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 16T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−9.005568796603216054148463994725, −8.576018554529311497264334828884, −7.29920281428733611690020109441, −6.93012124686086045475158498470, −6.41733731262631075275075007114, −5.31397458398503663343686473893, −4.27355983583208869261111283968, −3.30543849513860444509813513783, −2.23512148930571499620864151192, −1.16006993820417020799848605996,
0.33265120961226415085162446587, 2.37845823048201808574836704749, 3.25550939055484864900328375471, 3.90679468927351554000943454559, 5.17439977888095293148851471069, 5.67214574559406365429699901661, 6.36440104201510944507106353488, 7.69583774122624360742788065592, 8.286216706435420671037694790869, 9.140359473819741912045290429027
