| L(s) = 1 | + 3·3-s − 20·7-s + 9·9-s − 56·11-s + 86·13-s + 106·17-s + 4·19-s − 60·21-s − 136·23-s + 27·27-s − 206·29-s − 152·31-s − 168·33-s − 282·37-s + 258·39-s − 246·41-s − 412·43-s − 40·47-s + 57·49-s + 318·51-s + 126·53-s + 12·57-s + 56·59-s − 2·61-s − 180·63-s + 388·67-s − 408·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.07·7-s + 1/3·9-s − 1.53·11-s + 1.83·13-s + 1.51·17-s + 0.0482·19-s − 0.623·21-s − 1.23·23-s + 0.192·27-s − 1.31·29-s − 0.880·31-s − 0.886·33-s − 1.25·37-s + 1.05·39-s − 0.937·41-s − 1.46·43-s − 0.124·47-s + 0.166·49-s + 0.873·51-s + 0.326·53-s + 0.0278·57-s + 0.123·59-s − 0.00419·61-s − 0.359·63-s + 0.707·67-s − 0.711·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 56 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 106 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 136 T + p^{3} T^{2} \) |
| 29 | \( 1 + 206 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 282 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 40 T + p^{3} T^{2} \) |
| 53 | \( 1 - 126 T + p^{3} T^{2} \) |
| 59 | \( 1 - 56 T + p^{3} T^{2} \) |
| 61 | \( 1 + 2 T + p^{3} T^{2} \) |
| 67 | \( 1 - 388 T + p^{3} T^{2} \) |
| 71 | \( 1 + 672 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 - 408 T + p^{3} T^{2} \) |
| 83 | \( 1 + 668 T + p^{3} T^{2} \) |
| 89 | \( 1 - 66 T + p^{3} T^{2} \) |
| 97 | \( 1 - 926 T + p^{3} T^{2} \) |
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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
−10.00949230384556564167374542793, −8.864320722600307491350582166437, −8.117285377954784462420338163054, −7.28748501609370505265861103468, −6.09605233773722902826135059366, −5.36731941521076290106281303232, −3.67323308812194972118082389671, −3.21716071887988136811979169068, −1.71878145410417427398019709314, 0,
1.71878145410417427398019709314, 3.21716071887988136811979169068, 3.67323308812194972118082389671, 5.36731941521076290106281303232, 6.09605233773722902826135059366, 7.28748501609370505265861103468, 8.117285377954784462420338163054, 8.864320722600307491350582166437, 10.00949230384556564167374542793
