| L(s) = 1 | + 3.81·2-s − 9.32·3-s + 6.53·4-s − 5·5-s − 35.5·6-s + 25.5·7-s − 5.60·8-s + 59.8·9-s − 19.0·10-s + 11·11-s − 60.8·12-s − 31.4·13-s + 97.5·14-s + 46.6·15-s − 73.5·16-s + 2.16·17-s + 228.·18-s + 19·19-s − 32.6·20-s − 238.·21-s + 41.9·22-s − 121.·23-s + 52.2·24-s + 25·25-s − 119.·26-s − 306.·27-s + 167.·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 1.79·3-s + 0.816·4-s − 0.447·5-s − 2.41·6-s + 1.38·7-s − 0.247·8-s + 2.21·9-s − 0.602·10-s + 0.301·11-s − 1.46·12-s − 0.670·13-s + 1.86·14-s + 0.802·15-s − 1.14·16-s + 0.0308·17-s + 2.98·18-s + 0.229·19-s − 0.365·20-s − 2.47·21-s + 0.406·22-s − 1.10·23-s + 0.443·24-s + 0.200·25-s − 0.903·26-s − 2.18·27-s + 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.036138816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036138816\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 3.81T + 8T^{2} \) |
| 3 | \( 1 + 9.32T + 27T^{2} \) |
| 7 | \( 1 - 25.5T + 343T^{2} \) |
| 13 | \( 1 + 31.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.16T + 4.91e3T^{2} \) |
| 23 | \( 1 + 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 158.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 18.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 26.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 493.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 45.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 843.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 796.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 858.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 544.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 439.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 17.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 698.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−9.939213926220354605245572134633, −8.574008925080194443287670663804, −7.43725449900596321743674676314, −6.71482118647987292882517917479, −5.72934189749242079266020741582, −5.17919589770137286144928854681, −4.53388875274836988927075948853, −3.84409408677022880321620486277, −2.07721782981469509573950869937, −0.67580466476355945802924665055,
0.67580466476355945802924665055, 2.07721782981469509573950869937, 3.84409408677022880321620486277, 4.53388875274836988927075948853, 5.17919589770137286144928854681, 5.72934189749242079266020741582, 6.71482118647987292882517917479, 7.43725449900596321743674676314, 8.574008925080194443287670663804, 9.939213926220354605245572134633
