| L(s) = 1 | − 27·9-s + 18·13-s + 94·17-s − 130·29-s − 214·37-s − 230·41-s − 343·49-s − 518·53-s + 830·61-s − 1.09e3·73-s + 729·81-s − 1.67e3·89-s − 594·97-s + 598·101-s − 1.74e3·109-s − 2.00e3·113-s − 486·117-s + ⋯ |
| L(s) = 1 | − 9-s + 0.384·13-s + 1.34·17-s − 0.832·29-s − 0.950·37-s − 0.876·41-s − 49-s − 1.34·53-s + 1.74·61-s − 1.76·73-s + 81-s − 1.98·89-s − 0.621·97-s + 0.589·101-s − 1.53·109-s − 1.66·113-s − 0.384·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + 518 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 830 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + 1670 T + p^{3} T^{2} \) |
| 97 | \( 1 + 594 T + p^{3} T^{2} \) |
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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
−9.454062266051037699733296682746, −8.530586235852809458546550653502, −7.87792029467689850363567061215, −6.81571011651379039561216425778, −5.80404992327549873798859705001, −5.14816730863485631589480985998, −3.75273709825278212542140831976, −2.91242481392667512783582463272, −1.49365149367088566755810059792, 0,
1.49365149367088566755810059792, 2.91242481392667512783582463272, 3.75273709825278212542140831976, 5.14816730863485631589480985998, 5.80404992327549873798859705001, 6.81571011651379039561216425778, 7.87792029467689850363567061215, 8.530586235852809458546550653502, 9.454062266051037699733296682746
