| L(s) = 1 | − 2-s + 3.44·3-s + 4-s + 5-s − 3.44·6-s + 4.56·7-s − 8-s + 8.87·9-s − 10-s − 3.59·11-s + 3.44·12-s + 13-s − 4.56·14-s + 3.44·15-s + 16-s + 6.24·17-s − 8.87·18-s − 4.80·19-s + 20-s + 15.7·21-s + 3.59·22-s + 4.70·23-s − 3.44·24-s + 25-s − 26-s + 20.2·27-s + 4.56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.98·3-s + 0.5·4-s + 0.447·5-s − 1.40·6-s + 1.72·7-s − 0.353·8-s + 2.95·9-s − 0.316·10-s − 1.08·11-s + 0.994·12-s + 0.277·13-s − 1.21·14-s + 0.889·15-s + 0.250·16-s + 1.51·17-s − 2.09·18-s − 1.10·19-s + 0.223·20-s + 3.43·21-s + 0.766·22-s + 0.980·23-s − 0.703·24-s + 0.200·25-s − 0.196·26-s + 3.89·27-s + 0.862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.100170106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.100170106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 3.44T + 3T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 + 0.684T + 29T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 0.472T + 41T^{2} \) |
| 43 | \( 1 + 8.13T + 43T^{2} \) |
| 47 | \( 1 + 9.97T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 + 7.61T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 7.42T + 71T^{2} \) |
| 73 | \( 1 + 1.88T + 73T^{2} \) |
| 79 | \( 1 + 7.82T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 7.29T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.462570243278955274341581555572, −7.87272723622528789338142991605, −7.53826618607289713600733553983, −6.61034960192877514696747223514, −5.24065107050503317484038038932, −4.67163533158459148425442377255, −3.47933512419158304772058174143, −2.78831668224305435342473616046, −1.81671994958684817299811978845, −1.43863608761942804751420509787,
1.43863608761942804751420509787, 1.81671994958684817299811978845, 2.78831668224305435342473616046, 3.47933512419158304772058174143, 4.67163533158459148425442377255, 5.24065107050503317484038038932, 6.61034960192877514696747223514, 7.53826618607289713600733553983, 7.87272723622528789338142991605, 8.462570243278955274341581555572
