| L(s) = 1 | + 3.55·3-s + 41.6·5-s − 230.·9-s − 110.·11-s − 179.·13-s + 148.·15-s + 355.·17-s + 1.95e3·19-s − 1.54e3·23-s − 1.39e3·25-s − 1.68e3·27-s + 6.27e3·29-s + 6.00e3·31-s − 394.·33-s − 9.68e3·37-s − 637.·39-s − 1.05e4·41-s − 6.71e3·43-s − 9.59e3·45-s + 2.72e4·47-s + 1.26e3·51-s − 3.26e4·53-s − 4.61e3·55-s + 6.96e3·57-s − 492.·59-s − 4.05e4·61-s − 7.45e3·65-s + ⋯ |
| L(s) = 1 | + 0.228·3-s + 0.744·5-s − 0.947·9-s − 0.275·11-s − 0.293·13-s + 0.170·15-s + 0.298·17-s + 1.24·19-s − 0.609·23-s − 0.445·25-s − 0.444·27-s + 1.38·29-s + 1.12·31-s − 0.0630·33-s − 1.16·37-s − 0.0671·39-s − 0.982·41-s − 0.553·43-s − 0.706·45-s + 1.79·47-s + 0.0681·51-s − 1.59·53-s − 0.205·55-s + 0.283·57-s − 0.0184·59-s − 1.39·61-s − 0.218·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.55T + 243T^{2} \) |
| 5 | \( 1 - 41.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 110.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 179.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 355.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.95e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.71e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.72e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.26e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 492.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.68e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.39e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.57e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.85e4T + 8.58e9T^{2} \) |
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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
−9.139165226575988505527927282882, −8.304939440282318683813272879000, −7.49557710289560812454170785488, −6.33795826832038386607955291570, −5.61118146911858915184983210969, −4.75456554287692472853465079940, −3.31531713988108530776069415024, −2.55299351929761451792048409165, −1.37994708492961811316041756148, 0,
1.37994708492961811316041756148, 2.55299351929761451792048409165, 3.31531713988108530776069415024, 4.75456554287692472853465079940, 5.61118146911858915184983210969, 6.33795826832038386607955291570, 7.49557710289560812454170785488, 8.304939440282318683813272879000, 9.139165226575988505527927282882
