| L(s) = 1 | + 3·3-s + 4·5-s + 12·7-s + 9·9-s − 40·11-s − 40·13-s + 12·15-s − 66·17-s + 19·19-s + 36·21-s + 98·23-s − 109·25-s + 27·27-s − 130·29-s − 262·31-s − 120·33-s + 48·35-s − 296·37-s − 120·39-s − 442·41-s + 164·43-s + 36·45-s + 542·47-s − 199·49-s − 198·51-s + 334·53-s − 160·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.357·5-s + 0.647·7-s + 1/3·9-s − 1.09·11-s − 0.853·13-s + 0.206·15-s − 0.941·17-s + 0.229·19-s + 0.374·21-s + 0.888·23-s − 0.871·25-s + 0.192·27-s − 0.832·29-s − 1.51·31-s − 0.633·33-s + 0.231·35-s − 1.31·37-s − 0.492·39-s − 1.68·41-s + 0.581·43-s + 0.119·45-s + 1.68·47-s − 0.580·49-s − 0.543·51-s + 0.865·53-s − 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 40 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 23 | \( 1 - 98 T + p^{3} T^{2} \) |
| 29 | \( 1 + 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + 262 T + p^{3} T^{2} \) |
| 37 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 442 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 542 T + p^{3} T^{2} \) |
| 53 | \( 1 - 334 T + p^{3} T^{2} \) |
| 59 | \( 1 + 60 T + p^{3} T^{2} \) |
| 61 | \( 1 - 614 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + 400 T + p^{3} T^{2} \) |
| 73 | \( 1 - 318 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1154 T + p^{3} T^{2} \) |
| 83 | \( 1 - 636 T + p^{3} T^{2} \) |
| 89 | \( 1 + 630 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1006 T + p^{3} T^{2} \) |
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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
−9.197549221560276885279563357155, −8.525859545694449210429491088313, −7.53879250189483114796929676255, −7.02728577762577665241818946432, −5.55124509635269323926523230225, −4.96957166979774984662162464291, −3.78214563584989802655417688000, −2.54295855147646625853149554475, −1.78418164840669086357970975889, 0,
1.78418164840669086357970975889, 2.54295855147646625853149554475, 3.78214563584989802655417688000, 4.96957166979774984662162464291, 5.55124509635269323926523230225, 7.02728577762577665241818946432, 7.53879250189483114796929676255, 8.525859545694449210429491088313, 9.197549221560276885279563357155
