| L(s) = 1 | − 115.·2-s − 980.·3-s − 2.95e3·4-s + 1.17e5i·5-s + 1.13e5·6-s − 1.26e6i·7-s + 2.24e6·8-s − 3.82e6·9-s − 1.35e7i·10-s + 3.63e6i·11-s + 2.89e6·12-s − 1.04e8·13-s + 1.46e8i·14-s − 1.15e8i·15-s − 2.11e8·16-s − 6.51e8i·17-s + ⋯ |
| L(s) = 1 | − 0.905·2-s − 0.448·3-s − 0.180·4-s + 1.50i·5-s + 0.406·6-s − 1.53i·7-s + 1.06·8-s − 0.798·9-s − 1.35i·10-s + 0.186i·11-s + 0.0808·12-s − 1.66·13-s + 1.39i·14-s − 0.673i·15-s − 0.787·16-s − 1.58i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.4513352958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4513352958\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (2.61e9 - 2.17e9i)T \) |
| good | 2 | \( 1 + 115.T + 1.63e4T^{2} \) |
| 3 | \( 1 + 980.T + 4.78e6T^{2} \) |
| 5 | \( 1 - 1.17e5iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 1.26e6iT - 6.78e11T^{2} \) |
| 11 | \( 1 - 3.63e6iT - 3.79e14T^{2} \) |
| 13 | \( 1 + 1.04e8T + 3.93e15T^{2} \) |
| 17 | \( 1 + 6.51e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 4.41e8iT - 7.99e17T^{2} \) |
| 29 | \( 1 - 3.11e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 1.09e10T + 7.56e20T^{2} \) |
| 37 | \( 1 - 2.69e10iT - 9.01e21T^{2} \) |
| 41 | \( 1 - 7.90e10T + 3.79e22T^{2} \) |
| 43 | \( 1 - 3.19e11iT - 7.38e22T^{2} \) |
| 47 | \( 1 + 1.98e11T + 2.56e23T^{2} \) |
| 53 | \( 1 - 9.69e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 - 2.50e12T + 6.19e24T^{2} \) |
| 61 | \( 1 - 2.77e11iT - 9.87e24T^{2} \) |
| 67 | \( 1 - 1.37e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 + 1.25e13T + 8.27e25T^{2} \) |
| 73 | \( 1 + 1.16e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + 1.99e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 - 2.82e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 6.73e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 1.04e14iT - 6.52e27T^{2} \) |
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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
−14.42789901899512087157478853214, −13.80140856275446759275544145073, −11.57788827927437143723719528847, −10.49053974216955114695641536412, −9.723282137430728640033103556219, −7.62402599035394395253023530774, −6.90601386310989146572405495438, −4.68675213132811839812561693208, −2.78885467761497910518259656301, −0.58392210640830335534480917518,
0.40730889802701972248575083823, 2.02530519253977385363507192712, 4.73215964216823416286334282286, 5.75328379442833272410056703882, 8.298783464447994677494052669770, 8.779702331955077497434731022427, 10.08343935171734348000976684123, 12.01473999469839831681820801015, 12.64480908761917012616826186201, 14.52711906240379664694072724489
