L(s) = 1 | − 2-s − 2.99·3-s + 4-s − 3.60·5-s + 2.99·6-s + 0.0770·7-s − 8-s + 5.94·9-s + 3.60·10-s − 3.44·11-s − 2.99·12-s − 1.64·13-s − 0.0770·14-s + 10.7·15-s + 16-s + 1.47·17-s − 5.94·18-s − 4.21·19-s − 3.60·20-s − 0.230·21-s + 3.44·22-s − 23-s + 2.99·24-s + 7.96·25-s + 1.64·26-s − 8.79·27-s + 0.0770·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.72·3-s + 0.5·4-s − 1.61·5-s + 1.22·6-s + 0.0291·7-s − 0.353·8-s + 1.98·9-s + 1.13·10-s − 1.03·11-s − 0.863·12-s − 0.454·13-s − 0.0205·14-s + 2.78·15-s + 0.250·16-s + 0.357·17-s − 1.40·18-s − 0.967·19-s − 0.805·20-s − 0.0502·21-s + 0.734·22-s − 0.208·23-s + 0.610·24-s + 1.59·25-s + 0.321·26-s − 1.69·27-s + 0.0145·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 7 | \( 1 - 0.0770T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + 7.91T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 - 4.71T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 3.15T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 - 1.16T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 9.37T + 79T^{2} \) |
| 83 | \( 1 + 1.82T + 83T^{2} \) |
| 89 | \( 1 + 5.14T + 89T^{2} \) |
| 97 | \( 1 - 9.78T + 97T^{2} \) |
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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
−7.48998406758065514728538329888, −7.24666347512330026105695806121, −6.40242205786589364317375441932, −5.58295561542327229555472333981, −4.93680388152435224606464062010, −4.21386339227520988370858039116, −3.34655807784161983342776576697, −2.03993019900064092348418328871, −0.66752973816918681782109332342, 0,
0.66752973816918681782109332342, 2.03993019900064092348418328871, 3.34655807784161983342776576697, 4.21386339227520988370858039116, 4.93680388152435224606464062010, 5.58295561542327229555472333981, 6.40242205786589364317375441932, 7.24666347512330026105695806121, 7.48998406758065514728538329888