| L(s) = 1 | − 1.84i·3-s − 0.765i·5-s − 4.46i·7-s − 0.414·9-s + 1.84i·11-s − 1.41·15-s + 6.24·19-s − 8.24·21-s − 4.90i·23-s + 4.41·25-s − 4.77i·27-s − 2.29i·29-s − 5.54i·31-s + 3.41·33-s − 3.41·35-s + ⋯ |
| L(s) = 1 | − 1.06i·3-s − 0.342i·5-s − 1.68i·7-s − 0.138·9-s + 0.557i·11-s − 0.365·15-s + 1.43·19-s − 1.79·21-s − 1.02i·23-s + 0.882·25-s − 0.919i·27-s − 0.426i·29-s − 0.995i·31-s + 0.594·33-s − 0.577·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.763416771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.763416771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 1.84iT - 3T^{2} \) |
| 5 | \( 1 + 0.765iT - 5T^{2} \) |
| 7 | \( 1 + 4.46iT - 7T^{2} \) |
| 11 | \( 1 - 1.84iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 + 4.90iT - 23T^{2} \) |
| 29 | \( 1 + 2.29iT - 29T^{2} \) |
| 31 | \( 1 + 5.54iT - 31T^{2} \) |
| 37 | \( 1 - 1.84iT - 37T^{2} \) |
| 41 | \( 1 + 9.68iT - 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 - 2.48iT - 71T^{2} \) |
| 73 | \( 1 - 14.0iT - 73T^{2} \) |
| 79 | \( 1 - 13.5iT - 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 - 2.03iT - 97T^{2} \) |
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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
−8.361699638942018944419862577609, −7.76785510976666045193960472224, −6.98533430132177888699340455263, −6.81456748594569811862952194597, −5.55628350875425261166124568881, −4.53752338596014859818246033808, −3.88593046390083072635768837373, −2.59915947265925990762667128948, −1.36291722421405065695845155632, −0.66195738697873842237529173407,
1.61355261222329820353182315669, 3.11697860514242253552606191941, 3.27400789438699735421586502755, 4.79236811647488140999428818976, 5.25091959417713687167516158780, 6.02624115327736124281743969712, 6.95217517408932309786864442775, 8.005733413383609408817621147592, 8.752992933994255810224283364205, 9.496954805749674171441270026962
