| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 4i·7-s − i·8-s − 9-s − 4·11-s − i·12-s + 2i·13-s + 4·14-s + 16-s − 2i·17-s − i·18-s − 4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s + 0.554i·13-s + 1.06·14-s + 0.250·16-s − 0.485i·17-s − 0.235i·18-s − 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080969779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080969779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.394022292481085227518729019548, −7.75475318147133919743262546715, −7.12948124301263205297912561495, −6.51048945565580877858949820364, −5.51165096256350137043810090187, −4.85697016481569259983874591109, −4.15747629675926135435545382868, −3.52132930027113168396251459676, −2.34288383281389928116130857484, −0.812648249560500697296726980955,
0.38996351335477283426337970550, 1.86814498157576651718195197531, 2.53050988853198620136101480404, 3.06142356309532815642618560495, 4.35348170402238053517335743892, 5.15851472978656081536133065093, 5.89798258113482087571171974761, 6.36120002142041072559448162399, 7.73997882823514993689838467110, 8.022034066296966564146756463756
