| L(s) = 1 | − i·2-s − 4-s + 0.335·5-s + (−2.35 + 1.21i)7-s + i·8-s − 0.335i·10-s + 0.185i·11-s + 0.757i·13-s + (1.21 + 2.35i)14-s + 16-s − 0.456·17-s − i·19-s − 0.335·20-s + 0.185·22-s + 1.64i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.149·5-s + (−0.889 + 0.457i)7-s + 0.353i·8-s − 0.106i·10-s + 0.0558i·11-s + 0.209i·13-s + (0.323 + 0.628i)14-s + 0.250·16-s − 0.110·17-s − 0.229i·19-s − 0.0749·20-s + 0.0394·22-s + 0.344i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.222715324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222715324\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.35 - 1.21i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.335T + 5T^{2} \) |
| 11 | \( 1 - 0.185iT - 11T^{2} \) |
| 13 | \( 1 - 0.757iT - 13T^{2} \) |
| 17 | \( 1 + 0.456T + 17T^{2} \) |
| 23 | \( 1 - 1.64iT - 23T^{2} \) |
| 29 | \( 1 + 3.90iT - 29T^{2} \) |
| 31 | \( 1 + 9.21iT - 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 5.93T + 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 - 4.49iT - 53T^{2} \) |
| 59 | \( 1 + 6.16T + 59T^{2} \) |
| 61 | \( 1 + 4.83iT - 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 5.10iT - 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 0.519T + 79T^{2} \) |
| 83 | \( 1 + 8.81T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 + 16.2iT - 97T^{2} \) |
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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
−9.153896108632118431281695533022, −8.029739875826726971494934485083, −7.35637025233495998074026908666, −6.08896407853453438658090659829, −5.84260722763897908181032011847, −4.52292691913765118502622220364, −3.83415315107174406086996342788, −2.76780878020860300232871284868, −2.06756055587579420842823065205, −0.51084845386898621225981509869,
0.975860479366878436994642999968, 2.59098367900698328805426155849, 3.62532092993358217378024621989, 4.38852824700748252366664885972, 5.45666537646277859132277128002, 6.12110587163224442600679132396, 6.87162876481071481104946224125, 7.51452554878737376752513772734, 8.341214616282868042184769506160, 9.164708823858847294245262874919
