L(s) = 1 | + 5·2-s + 17·4-s + 45·8-s − 27·9-s − 68·11-s + 89·16-s − 135·18-s − 340·22-s + 40·23-s − 166·29-s + 85·32-s − 459·36-s − 450·37-s + 180·43-s − 1.15e3·44-s + 200·46-s − 590·53-s − 830·58-s − 287·64-s + 740·67-s + 688·71-s − 1.21e3·72-s − 2.25e3·74-s − 1.38e3·79-s + 729·81-s + 900·86-s − 3.06e3·88-s + ⋯ |
L(s) = 1 | + 1.76·2-s + 17/8·4-s + 1.98·8-s − 9-s − 1.86·11-s + 1.39·16-s − 1.76·18-s − 3.29·22-s + 0.362·23-s − 1.06·29-s + 0.469·32-s − 2.12·36-s − 1.99·37-s + 0.638·43-s − 3.96·44-s + 0.641·46-s − 1.52·53-s − 1.87·58-s − 0.560·64-s + 1.34·67-s + 1.15·71-s − 1.98·72-s − 3.53·74-s − 1.97·79-s + 81-s + 1.12·86-s − 3.70·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 3 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 68 T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 - 40 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 450 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 180 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + 590 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + p^{3} T^{2} \) |
| 67 | \( 1 - 740 T + p^{3} T^{2} \) |
| 71 | \( 1 - 688 T + p^{3} T^{2} \) |
| 73 | \( 1 + p^{3} T^{2} \) |
| 79 | \( 1 + 1384 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + p^{3} T^{2} \) |
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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
−8.784853078304363255888346571584, −7.85042915357328648786477403234, −7.08012648942760879408187945070, −6.03074505577547228334097971392, −5.35839426906832944408285085560, −4.87919502475217459673920068436, −3.60814427670431172414485522132, −2.88453729935879048586939345379, −2.06407466010492228484826349351, 0,
2.06407466010492228484826349351, 2.88453729935879048586939345379, 3.60814427670431172414485522132, 4.87919502475217459673920068436, 5.35839426906832944408285085560, 6.03074505577547228334097971392, 7.08012648942760879408187945070, 7.85042915357328648786477403234, 8.784853078304363255888346571584