| L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 3·9-s − 2·10-s − 4·11-s + 6·13-s + 14-s + 16-s − 6·17-s − 3·18-s − 4·19-s − 2·20-s − 4·22-s − 25-s + 6·26-s + 28-s + 29-s − 4·31-s + 32-s − 6·34-s − 2·35-s − 3·36-s − 6·37-s − 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.632·10-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 1.17·26-s + 0.188·28-s + 0.185·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s − 1/2·36-s − 0.986·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214774 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214774 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.25458463158386, −12.73202243470962, −12.45324429647209, −11.60694216976558, −11.48285023868995, −10.93161316288953, −10.70079065405408, −10.32446435011027, −9.224600819988631, −8.916387465084665, −8.293205117518331, −8.127270521262668, −7.641980225299490, −6.790543647755397, −6.589300811405333, −5.857408071407653, −5.531294004203871, −4.888722462110010, −4.437199924474199, −3.814210740352360, −3.481916751567585, −2.846516324449024, −2.167062736944059, −1.771828769893423, −0.6486846023569878, 0,
0.6486846023569878, 1.771828769893423, 2.167062736944059, 2.846516324449024, 3.481916751567585, 3.814210740352360, 4.437199924474199, 4.888722462110010, 5.531294004203871, 5.857408071407653, 6.589300811405333, 6.790543647755397, 7.641980225299490, 8.127270521262668, 8.293205117518331, 8.916387465084665, 9.224600819988631, 10.32446435011027, 10.70079065405408, 10.93161316288953, 11.48285023868995, 11.60694216976558, 12.45324429647209, 12.73202243470962, 13.25458463158386
