| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 4.82·7-s − 8-s + 9-s − 10-s − 2·11-s + 12-s − 3.41·13-s + 4.82·14-s + 15-s + 16-s − 18-s + 2.82·19-s + 20-s − 4.82·21-s + 2·22-s + 2.24·23-s − 24-s + 25-s + 3.41·26-s + 27-s − 4.82·28-s + 1.17·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.82·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.946·13-s + 1.29·14-s + 0.258·15-s + 0.250·16-s − 0.235·18-s + 0.648·19-s + 0.223·20-s − 1.05·21-s + 0.426·22-s + 0.467·23-s − 0.204·24-s + 0.200·25-s + 0.669·26-s + 0.192·27-s − 0.912·28-s + 0.217·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.34T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.51578219827997941523321285337, −6.82366691258847311403365518443, −6.26303510315235787914477183656, −5.54200951837380319838614192830, −4.59911081168909260913848296382, −3.55454062369331617128653585224, −2.71107207938432249451679412079, −2.55857421669886346023351119024, −1.11501148453165171991849050481, 0,
1.11501148453165171991849050481, 2.55857421669886346023351119024, 2.71107207938432249451679412079, 3.55454062369331617128653585224, 4.59911081168909260913848296382, 5.54200951837380319838614192830, 6.26303510315235787914477183656, 6.82366691258847311403365518443, 7.51578219827997941523321285337
