| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 11-s − 12-s − 2·14-s + 16-s + 17-s − 18-s − 2·21-s − 22-s + 6·23-s + 24-s − 27-s + 2·28-s − 2·29-s + 4·31-s − 32-s − 33-s − 34-s + 36-s − 2·37-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.436·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s − 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.174·33-s − 0.171·34-s + 1/6·36-s − 0.328·37-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.67860752360587, −14.95397316290196, −14.61022563490295, −13.94657244938796, −13.30710613073045, −12.68934758492790, −12.12259301062616, −11.59713747677358, −11.17777801838619, −10.67867966997223, −10.13187854571693, −9.532845992234303, −8.913086674771363, −8.440314633853248, −7.761427684363857, −7.238207092666710, −6.695747240676085, −6.042730770375963, −5.377921626400467, −4.823545920983408, −4.146262547518159, −3.271555391922100, −2.523296862999023, −1.554235238172732, −1.095712010298483, 0,
1.095712010298483, 1.554235238172732, 2.523296862999023, 3.271555391922100, 4.146262547518159, 4.823545920983408, 5.377921626400467, 6.042730770375963, 6.695747240676085, 7.238207092666710, 7.761427684363857, 8.440314633853248, 8.913086674771363, 9.532845992234303, 10.13187854571693, 10.67867966997223, 11.17777801838619, 11.59713747677358, 12.12259301062616, 12.68934758492790, 13.30710613073045, 13.94657244938796, 14.61022563490295, 14.95397316290196, 15.67860752360587
