| L(s) = 1 | − 5.97·3-s + 11.6·5-s − 8.29·7-s + 8.66·9-s − 49.0·11-s + 24.9·13-s − 69.2·15-s − 42.7·17-s − 146.·19-s + 49.5·21-s − 12.9·23-s + 9.57·25-s + 109.·27-s − 29·29-s − 133.·31-s + 293.·33-s − 96.2·35-s + 381.·37-s − 149.·39-s − 416.·41-s − 447.·43-s + 100.·45-s + 555.·47-s − 274.·49-s + 255.·51-s + 308.·53-s − 569.·55-s + ⋯ |
| L(s) = 1 | − 1.14·3-s + 1.03·5-s − 0.447·7-s + 0.321·9-s − 1.34·11-s + 0.533·13-s − 1.19·15-s − 0.610·17-s − 1.77·19-s + 0.514·21-s − 0.117·23-s + 0.0765·25-s + 0.780·27-s − 0.185·29-s − 0.775·31-s + 1.54·33-s − 0.464·35-s + 1.69·37-s − 0.612·39-s − 1.58·41-s − 1.58·43-s + 0.333·45-s + 1.72·47-s − 0.799·49-s + 0.701·51-s + 0.800·53-s − 1.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6835767374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6835767374\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 5.97T + 27T^{2} \) |
| 5 | \( 1 - 11.6T + 125T^{2} \) |
| 7 | \( 1 + 8.29T + 343T^{2} \) |
| 11 | \( 1 + 49.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 12.9T + 1.21e4T^{2} \) |
| 31 | \( 1 + 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 381.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 416.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 447.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 555.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 308.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 647.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 394.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 216.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 408.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 218.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 928.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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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
−8.867376458985717443421145624074, −8.219743253498888546049367051353, −6.96665639393251124094438898006, −6.31282304093805909604906227586, −5.72628390001580829470916168983, −5.11191124701775321477163234507, −4.09036500835493654756380824588, −2.70876647552423500717866031368, −1.87018129696503953238485085770, −0.39042925884778436420694122851,
0.39042925884778436420694122851, 1.87018129696503953238485085770, 2.70876647552423500717866031368, 4.09036500835493654756380824588, 5.11191124701775321477163234507, 5.72628390001580829470916168983, 6.31282304093805909604906227586, 6.96665639393251124094438898006, 8.219743253498888546049367051353, 8.867376458985717443421145624074
