| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 12-s − 13-s − 14-s + 15-s + 16-s + 2·17-s − 18-s − 19-s − 20-s − 21-s + 6·23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9616778313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9616778313\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.57137368680491, −14.21891101399247, −13.19249448366729, −12.84092972671878, −12.44624767577028, −11.65022690979601, −11.30907387727796, −11.03822978155822, −10.38725944154134, −9.788052059551715, −9.326367939635224, −8.805627787471036, −8.015113404449225, −7.764008793034319, −7.134504095218250, −6.646292922078989, −5.954158033272329, −5.417709026130159, −4.747066382410125, −4.235453715900604, −3.366725872792290, −2.803337802727986, −1.885117692405260, −1.221292529518161, −0.4357400790827405,
0.4357400790827405, 1.221292529518161, 1.885117692405260, 2.803337802727986, 3.366725872792290, 4.235453715900604, 4.747066382410125, 5.417709026130159, 5.954158033272329, 6.646292922078989, 7.134504095218250, 7.764008793034319, 8.015113404449225, 8.805627787471036, 9.326367939635224, 9.788052059551715, 10.38725944154134, 11.03822978155822, 11.30907387727796, 11.65022690979601, 12.44624767577028, 12.84092972671878, 13.19249448366729, 14.21891101399247, 14.57137368680491
