| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 0.307·7-s + 8-s + 9-s − 10-s − 0.335·11-s + 12-s − 0.307·14-s − 15-s + 16-s − 6.85·17-s + 18-s − 5.80·19-s − 20-s − 0.307·21-s − 0.335·22-s + 2.35·23-s + 24-s + 25-s + 27-s − 0.307·28-s − 5.91·29-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 − 0.116·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.101·11-s + 0.288·12-s − 0.0823·14-s − 0.258·15-s + 0.250·16-s − 1.66·17-s + 0.235·18-s − 1.33·19-s − 0.223·20-s − 0.0672·21-s − 0.0714·22-s + 0.491·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.0582·28-s − 1.09·29-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 0.307T + 7T^{2} \) |
| 11 | \( 1 + 0.335T + 11T^{2} \) |
| 17 | \( 1 + 6.85T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 + 0.0609T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 3.40T + 41T^{2} \) |
| 43 | \( 1 + 0.0489T + 43T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 2.14T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 9.56T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 + 0.0760T + 79T^{2} \) |
| 83 | \( 1 + 3.84T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 + 6.13T + 97T^{2} \) |
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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
−7.82969542147301176358727109275, −6.94368080344642582282784024240, −6.64265731929009121934322166749, −5.61020763371688111555033428855, −4.72198793601738947029828679471, −4.12471441616175329672759378216, −3.42175719230059688031750651754, −2.48045348851859066362084186641, −1.75294539356645044996079501842, 0,
1.75294539356645044996079501842, 2.48045348851859066362084186641, 3.42175719230059688031750651754, 4.12471441616175329672759378216, 4.72198793601738947029828679471, 5.61020763371688111555033428855, 6.64265731929009121934322166749, 6.94368080344642582282784024240, 7.82969542147301176358727109275
