| L(s) = 1 | + 4.53·2-s + 3.57·3-s + 12.5·4-s − 13.4·5-s + 16.2·6-s + 20.5·8-s − 14.1·9-s − 61.0·10-s + 0.813·11-s + 44.8·12-s + 34.9·13-s − 48.2·15-s − 7.21·16-s + 117.·17-s − 64.2·18-s − 93.2·19-s − 168.·20-s + 3.68·22-s + 120.·23-s + 73.4·24-s + 56.6·25-s + 158.·26-s − 147.·27-s + 8.56·29-s − 218.·30-s − 82.1·31-s − 196.·32-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.688·3-s + 1.56·4-s − 1.20·5-s + 1.10·6-s + 0.907·8-s − 0.525·9-s − 1.93·10-s + 0.0222·11-s + 1.07·12-s + 0.745·13-s − 0.830·15-s − 0.112·16-s + 1.67·17-s − 0.841·18-s − 1.12·19-s − 1.88·20-s + 0.0357·22-s + 1.09·23-s + 0.625·24-s + 0.453·25-s + 1.19·26-s − 1.05·27-s + 0.0548·29-s − 1.33·30-s − 0.475·31-s − 1.08·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.957252836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957252836\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 4.53T + 8T^{2} \) |
| 3 | \( 1 - 3.57T + 27T^{2} \) |
| 5 | \( 1 + 13.4T + 125T^{2} \) |
| 11 | \( 1 - 0.813T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.56T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 28.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 70.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 417.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 94.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 120.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 792.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 104.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 550.T + 9.12e5T^{2} \) |
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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
−14.85709300458867572323408325636, −14.11934434124373552550574324532, −12.92233742036783344927795430096, −11.95469959029108008654121383700, −10.95577004290309420428988011852, −8.742629377184811453903417643264, −7.46755154542505827323762836077, −5.77036537804287203175257888116, −4.09533863556181174331652822198, −3.07853439106705805267098237945,
3.07853439106705805267098237945, 4.09533863556181174331652822198, 5.77036537804287203175257888116, 7.46755154542505827323762836077, 8.742629377184811453903417643264, 10.95577004290309420428988011852, 11.95469959029108008654121383700, 12.92233742036783344927795430096, 14.11934434124373552550574324532, 14.85709300458867572323408325636
