| L(s) = 1 | + i·3-s + 1.33·7-s − 9-s − 2.94i·11-s + 2.04i·13-s − 3.61·17-s − 5.35i·19-s + 1.33i·21-s + 8.59·23-s − i·27-s + 5.26i·29-s + 2.08·31-s + 2.94·33-s − 6.55i·37-s − 2.04·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.504·7-s − 0.333·9-s − 0.887i·11-s + 0.566i·13-s − 0.876·17-s − 1.22i·19-s + 0.291i·21-s + 1.79·23-s − 0.192i·27-s + 0.977i·29-s + 0.373·31-s + 0.512·33-s − 1.07i·37-s − 0.326·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.834992868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834992868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 + 2.94iT - 11T^{2} \) |
| 13 | \( 1 - 2.04iT - 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + 5.35iT - 19T^{2} \) |
| 23 | \( 1 - 8.59T + 23T^{2} \) |
| 29 | \( 1 - 5.26iT - 29T^{2} \) |
| 31 | \( 1 - 2.08T + 31T^{2} \) |
| 37 | \( 1 + 6.55iT - 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 + 8.50iT - 43T^{2} \) |
| 47 | \( 1 - 9.97T + 47T^{2} \) |
| 53 | \( 1 - 6.12iT - 53T^{2} \) |
| 59 | \( 1 + 4.75iT - 59T^{2} \) |
| 61 | \( 1 + 8.51iT - 61T^{2} \) |
| 67 | \( 1 - 10.6iT - 67T^{2} \) |
| 71 | \( 1 - 2.62T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 1.52iT - 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.943551098487152746120641118029, −8.504337920807420811237858772848, −7.30828428171260139059383031805, −6.73830519957265639523379950960, −5.67712185286213532846441326591, −4.93637059440398814327811985359, −4.23681628851993973859432913531, −3.20900945751861459016632135891, −2.28794147385858847928061816070, −0.77229503198967958741467975438,
1.02920738849212541195631510185, 2.09316875225004068097589727520, 3.02697639235422243873275710958, 4.27471331702123372560713694295, 4.98011400043986088599332146190, 5.94175952917961954366968901574, 6.70141819584204730686622655704, 7.52519832861648644086109874349, 8.058802115099872870571414807259, 8.869195307629429127378918641464
