| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 1.35·7-s + 8-s + 9-s + 10-s − 5.29·11-s − 12-s − 1.35·14-s − 15-s + 16-s − 2.04·17-s + 18-s + 3.24·19-s + 20-s + 1.35·21-s − 5.29·22-s + 1.33·23-s − 24-s + 25-s − 27-s − 1.35·28-s − 5.13·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.512·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.59·11-s − 0.288·12-s − 0.362·14-s − 0.258·15-s + 0.250·16-s − 0.496·17-s + 0.235·18-s + 0.744·19-s + 0.223·20-s + 0.296·21-s − 1.12·22-s + 0.278·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.256·28-s − 0.953·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.203044149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.203044149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 - 1.33T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 + 2.24T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 3.62T + 61T^{2} \) |
| 67 | \( 1 - 9.03T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 0.259T + 79T^{2} \) |
| 83 | \( 1 - 3.02T + 83T^{2} \) |
| 89 | \( 1 + 5.47T + 89T^{2} \) |
| 97 | \( 1 + 5.85T + 97T^{2} \) |
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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
−8.018856590583575090081154943004, −7.29840251424032216439573629048, −6.73594369238836365707173734377, −5.67846028066303987411594358665, −5.56530387514592173957019746677, −4.68379685580992027078147465904, −3.81195017545511677045912165084, −2.80445705041291362075024464616, −2.18118109815535274438921371647, −0.72585712268704062390585265377,
0.72585712268704062390585265377, 2.18118109815535274438921371647, 2.80445705041291362075024464616, 3.81195017545511677045912165084, 4.68379685580992027078147465904, 5.56530387514592173957019746677, 5.67846028066303987411594358665, 6.73594369238836365707173734377, 7.29840251424032216439573629048, 8.018856590583575090081154943004
