| L(s) = 1 | + (−1.58 − 0.707i)3-s + 5-s + 1.33i·7-s + (1.99 + 2.23i)9-s − 2.74·11-s − 4.49·13-s + (−1.58 − 0.707i)15-s + 3.35·17-s − 3.68i·19-s + (0.946 − 2.11i)21-s + (0.364 − 4.78i)23-s + 25-s + (−1.57 − 4.95i)27-s + 3.60i·29-s − 0.848·31-s + ⋯ |
| L(s) = 1 | + (−0.912 − 0.408i)3-s + 0.447·5-s + 0.505i·7-s + (0.666 + 0.745i)9-s − 0.829·11-s − 1.24·13-s + (−0.408 − 0.182i)15-s + 0.813·17-s − 0.844i·19-s + (0.206 − 0.461i)21-s + (0.0759 − 0.997i)23-s + 0.200·25-s + (−0.303 − 0.952i)27-s + 0.668i·29-s − 0.152·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6867358837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6867358837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 + 0.707i)T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (-0.364 + 4.78i)T \) |
| good | 7 | \( 1 - 1.33iT - 7T^{2} \) |
| 11 | \( 1 + 2.74T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 3.35T + 17T^{2} \) |
| 19 | \( 1 + 3.68iT - 19T^{2} \) |
| 29 | \( 1 - 3.60iT - 29T^{2} \) |
| 31 | \( 1 + 0.848T + 31T^{2} \) |
| 37 | \( 1 + 2.13iT - 37T^{2} \) |
| 41 | \( 1 + 5.16iT - 41T^{2} \) |
| 43 | \( 1 + 12.2iT - 43T^{2} \) |
| 47 | \( 1 - 0.193iT - 47T^{2} \) |
| 53 | \( 1 + 5.60T + 53T^{2} \) |
| 59 | \( 1 - 6.74iT - 59T^{2} \) |
| 61 | \( 1 + 1.26iT - 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 7.83iT - 71T^{2} \) |
| 73 | \( 1 + 6.09T + 73T^{2} \) |
| 79 | \( 1 + 16.9iT - 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 3.98T + 89T^{2} \) |
| 97 | \( 1 + 13.7iT - 97T^{2} \) |
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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
−9.406607609323005193706359239562, −8.499269526420441896263717176584, −7.42014965790013703679011800612, −6.96411843169197286434573342190, −5.81035060448969090072901603492, −5.29792610445214455776997980571, −4.53709785266006524961902855547, −2.85164424167858998248949257620, −1.97288324781133153244815648502, −0.32634558348411623336970752637,
1.30035767840170244980037393340, 2.81946779644711302962518995168, 4.00277548091391386267394014241, 4.98330874662894039922596958178, 5.56844213198944959095826681748, 6.43984719640781567347858481416, 7.43198051682265850155784155148, 8.009947085868439819353563579876, 9.522293095783417284655855985527, 9.857303165137899983753727891119
