| L(s) = 1 | + 1.56i·3-s + 1.56i·7-s + 0.561·9-s + 3.12·11-s − i·13-s − 6.68i·17-s + 3.12·19-s − 2.43·21-s + 8i·23-s + 5.56i·27-s + 2·29-s + 4·31-s + 4.87i·33-s − 2.68i·37-s + 1.56·39-s + ⋯ |
| L(s) = 1 | + 0.901i·3-s + 0.590i·7-s + 0.187·9-s + 0.941·11-s − 0.277i·13-s − 1.62i·17-s + 0.716·19-s − 0.532·21-s + 1.66i·23-s + 1.07i·27-s + 0.371·29-s + 0.718·31-s + 0.848i·33-s − 0.441i·37-s + 0.250·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.126110009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126110009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 - 1.56iT - 3T^{2} \) |
| 7 | \( 1 - 1.56iT - 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 17 | \( 1 + 6.68iT - 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2.68iT - 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 9.56iT - 43T^{2} \) |
| 47 | \( 1 + 12.6iT - 47T^{2} \) |
| 53 | \( 1 - 5.12iT - 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 - 3.12iT - 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 14.2iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8.24iT - 97T^{2} \) |
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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
−9.309532929611518832347427339824, −8.458512879497223167518975407116, −7.34179809100281504884500552410, −6.88478586710962422972596675418, −5.55408618802665455187914370301, −5.24024852743439684741035356903, −4.16059225580486580490814586628, −3.48825780368511270852995792402, −2.47272693607416979065566899386, −1.06841857355574265137674638716,
0.925220720116169909289406435641, 1.67116017240717675095663287501, 2.87078321645871810498035193460, 4.09874176045801630145908792133, 4.54038773955935742530112456956, 6.06224073123972797731578918310, 6.44696377189374238298877142814, 7.14995361239331762361560646178, 7.976923407647256566927889883732, 8.515698059564350467600321547635
