| L(s) = 1 | − 7.44·3-s + 16.9·5-s + 7·7-s + 28.3·9-s + 25.3·11-s − 33.5·13-s − 126.·15-s + 76.1·17-s + 78.3·19-s − 52.0·21-s + 125.·23-s + 162.·25-s − 10.0·27-s + 62.7·29-s − 48.0·31-s − 188.·33-s + 118.·35-s − 381.·37-s + 249.·39-s − 41·41-s + 548.·43-s + 480.·45-s − 85.2·47-s + 49·49-s − 566.·51-s − 307.·53-s + 429.·55-s + ⋯ |
| L(s) = 1 | − 1.43·3-s + 1.51·5-s + 0.377·7-s + 1.05·9-s + 0.694·11-s − 0.716·13-s − 2.16·15-s + 1.08·17-s + 0.945·19-s − 0.541·21-s + 1.14·23-s + 1.29·25-s − 0.0719·27-s + 0.401·29-s − 0.278·31-s − 0.994·33-s + 0.572·35-s − 1.69·37-s + 1.02·39-s − 0.156·41-s + 1.94·43-s + 1.59·45-s − 0.264·47-s + 0.142·49-s − 1.55·51-s − 0.797·53-s + 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.029727758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.029727758\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 + 7.44T + 27T^{2} \) |
| 5 | \( 1 - 16.9T + 125T^{2} \) |
| 11 | \( 1 - 25.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 78.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 62.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 48.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 381.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 548.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 85.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 165.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 310.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 397.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 568.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 10.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 357.T + 9.12e5T^{2} \) |
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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
−9.610263590400195901774229594866, −8.894002856646238114342017027387, −7.45528943023375329031335886077, −6.73908383676903245182997516986, −5.83175314908583974789184850611, −5.38308522280028198189906991207, −4.66488712157736733720256971746, −3.06919121712048697930216064971, −1.68255554108829049087920222233, −0.856672548705675130966810180439,
0.856672548705675130966810180439, 1.68255554108829049087920222233, 3.06919121712048697930216064971, 4.66488712157736733720256971746, 5.38308522280028198189906991207, 5.83175314908583974789184850611, 6.73908383676903245182997516986, 7.45528943023375329031335886077, 8.894002856646238114342017027387, 9.610263590400195901774229594866
