| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 5·5-s − 6·6-s + 8·8-s + 9·9-s + 10·10-s − 20·11-s − 12·12-s + 26·13-s − 15·15-s + 16·16-s − 26·17-s + 18·18-s − 42·19-s + 20·20-s − 40·22-s − 194·23-s − 24·24-s + 25·25-s + 52·26-s − 27·27-s − 42·29-s − 30·30-s + 274·31-s + 32·32-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 − 0.548·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.370·17-s + 0.235·18-s − 0.507·19-s + 0.223·20-s − 0.387·22-s − 1.75·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.268·29-s − 0.182·30-s + 1.58·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 26 T + p^{3} T^{2} \) |
| 19 | \( 1 + 42 T + p^{3} T^{2} \) |
| 23 | \( 1 + 194 T + p^{3} T^{2} \) |
| 29 | \( 1 + 42 T + p^{3} T^{2} \) |
| 31 | \( 1 - 274 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 + 296 T + p^{3} T^{2} \) |
| 47 | \( 1 + 328 T + p^{3} T^{2} \) |
| 53 | \( 1 + 148 T + p^{3} T^{2} \) |
| 59 | \( 1 - 488 T + p^{3} T^{2} \) |
| 61 | \( 1 - 272 T + p^{3} T^{2} \) |
| 67 | \( 1 - 8 T + p^{3} T^{2} \) |
| 71 | \( 1 + 684 T + p^{3} T^{2} \) |
| 73 | \( 1 - 310 T + p^{3} T^{2} \) |
| 79 | \( 1 + 584 T + p^{3} T^{2} \) |
| 83 | \( 1 - 404 T + p^{3} T^{2} \) |
| 89 | \( 1 + 266 T + p^{3} T^{2} \) |
| 97 | \( 1 + 678 T + p^{3} T^{2} \) |
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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
−8.578903436481942226926324290959, −7.914898057907616908896031280380, −6.72159208634528876655798592065, −6.23488910478472656097781138956, −5.42200574988281668953995898669, −4.61204559844536201543188454935, −3.71672829546453671200785105353, −2.51110606497287425723166793907, −1.52159850327207170643010383522, 0,
1.52159850327207170643010383522, 2.51110606497287425723166793907, 3.71672829546453671200785105353, 4.61204559844536201543188454935, 5.42200574988281668953995898669, 6.23488910478472656097781138956, 6.72159208634528876655798592065, 7.914898057907616908896031280380, 8.578903436481942226926324290959
