| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 10-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 20-s − 21-s + 24-s + 26-s + 27-s + 28-s + 30-s − 2·31-s − 32-s + 34-s + 35-s + 37-s + 39-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 10-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 20-s − 21-s + 24-s + 26-s + 27-s + 28-s + 30-s − 2·31-s − 32-s + 34-s + 35-s + 37-s + 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3506445621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3506445621\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−14.21903403686422288216786219204, −12.69759417877772506687250298073, −11.53875891143008359430010850022, −10.91362787156601425843494571816, −9.849520919914033047471247341525, −8.768893274904782034971084298372, −7.37748796465981917417358202634, −6.13973319085271586429221786246, −5.11000336544565001207901214497, −2.09431808749519266596519118860,
2.09431808749519266596519118860, 5.11000336544565001207901214497, 6.13973319085271586429221786246, 7.37748796465981917417358202634, 8.768893274904782034971084298372, 9.849520919914033047471247341525, 10.91362787156601425843494571816, 11.53875891143008359430010850022, 12.69759417877772506687250298073, 14.21903403686422288216786219204
