L(s) = 1 | − 0.000359·2-s − 7.63·3-s − 7.99·4-s + 5·5-s + 0.00274·6-s − 26.4·7-s + 0.00574·8-s + 31.3·9-s − 0.00179·10-s + 11·11-s + 61.0·12-s − 29.0·13-s + 0.00949·14-s − 38.1·15-s + 63.9·16-s − 65.6·17-s − 0.0112·18-s + 19·19-s − 39.9·20-s + 201.·21-s − 0.00395·22-s − 176.·23-s − 0.0438·24-s + 25·25-s + 0.0104·26-s − 33.0·27-s + 211.·28-s + ⋯ |
L(s) = 1 | − 0.000126·2-s − 1.46·3-s − 0.999·4-s + 0.447·5-s + 0.000186·6-s − 1.42·7-s + 0.000253·8-s + 1.16·9-s − 5.67e − 5·10-s + 0.301·11-s + 1.46·12-s − 0.619·13-s + 0.000181·14-s − 0.657·15-s + 0.999·16-s − 0.936·17-s − 0.000147·18-s + 0.229·19-s − 0.447·20-s + 2.09·21-s − 3.82e − 5·22-s − 1.60·23-s − 0.000373·24-s + 0.200·25-s + 7.86e−5·26-s − 0.235·27-s + 1.42·28-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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 + 0.000359T + 8T^{2} \) |
| 3 | \( 1 + 7.63T + 27T^{2} \) |
| 7 | \( 1 + 26.4T + 343T^{2} \) |
| 13 | \( 1 + 29.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.6T + 4.91e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 233.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 172.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 612.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 346.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 168.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 702.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 675.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 645.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 967.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 236.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 675.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 690.T + 9.12e5T^{2} \) |
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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
−9.376771008199200723353920829555, −8.476845998574844195602380543358, −7.14908044292721937995804070181, −6.22371013412712115482965656783, −5.90879796179087015792796250691, −4.77522260730632446897426176327, −4.11471111637507300046865768153, −2.70210729355105462641155818113, −0.860840580904576960409155344707, 0,
0.860840580904576960409155344707, 2.70210729355105462641155818113, 4.11471111637507300046865768153, 4.77522260730632446897426176327, 5.90879796179087015792796250691, 6.22371013412712115482965656783, 7.14908044292721937995804070181, 8.476845998574844195602380543358, 9.376771008199200723353920829555