| L(s) = 1 | − i·2-s − 3.11i·3-s − 4-s − 3.11·6-s − 4.50i·7-s + i·8-s − 6.72·9-s + 4.33·11-s + 3.11i·12-s − 3.72i·13-s − 4.50·14-s + 16-s − 1.11i·17-s + 6.72i·18-s − 4.50·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.80i·3-s − 0.5·4-s − 1.27·6-s − 1.70i·7-s + 0.353i·8-s − 2.24·9-s + 1.30·11-s + 0.900i·12-s − 1.03i·13-s − 1.20·14-s + 0.250·16-s − 0.271i·17-s + 1.58i·18-s − 1.03·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.400222910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400222910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 3.11iT - 3T^{2} \) |
| 7 | \( 1 + 4.50iT - 7T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 13 | \( 1 + 3.72iT - 13T^{2} \) |
| 17 | \( 1 + 1.11iT - 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 29 | \( 1 - 8.23T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 - 0.781iT - 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 + 2.43iT - 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + 9.45iT - 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 2.78iT - 83T^{2} \) |
| 89 | \( 1 - 7.69T + 89T^{2} \) |
| 97 | \( 1 - 0.642iT - 97T^{2} \) |
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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
−9.211290600947580527114849636775, −8.131098361610426389687239049113, −7.73375514506084037334009387410, −6.62480499318116692063902711901, −6.35064216412340256307899954303, −4.76459591049384446159368323857, −3.70426286237112574043637927993, −2.62031034995037913229378134318, −1.29316392511902349318088390131, −0.69254865095357059177978443704,
2.33734163032712120420745458619, 3.67210698886093732212753899882, 4.38582424447294994261595384175, 5.21094734394524584836971853641, 6.04019373163019993926386700090, 6.69595152108026386252218999609, 8.531121923300378097911099464731, 8.694567216881749317976809887821, 9.373540782287521625820071885044, 10.02933685703315855552205359540
