| L(s) = 1 | − 29.3·3-s + 64.0·5-s + 138.·7-s + 621.·9-s − 557.·11-s − 41.1·13-s − 1.88e3·15-s − 1.64e3·17-s − 258.·19-s − 4.08e3·21-s + 2.82e3·23-s + 981.·25-s − 1.11e4·27-s + 841·29-s + 5.98e3·31-s + 1.63e4·33-s + 8.89e3·35-s − 3.32e3·37-s + 1.21e3·39-s − 3.89e3·41-s + 3.58e3·43-s + 3.98e4·45-s + 7.50e3·47-s + 2.45e3·49-s + 4.83e4·51-s + 9.01e3·53-s − 3.57e4·55-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 1.14·5-s + 1.07·7-s + 2.55·9-s − 1.38·11-s − 0.0675·13-s − 2.16·15-s − 1.37·17-s − 0.164·19-s − 2.01·21-s + 1.11·23-s + 0.314·25-s − 2.93·27-s + 0.185·29-s + 1.11·31-s + 2.61·33-s + 1.22·35-s − 0.399·37-s + 0.127·39-s − 0.361·41-s + 0.296·43-s + 2.93·45-s + 0.495·47-s + 0.146·49-s + 2.60·51-s + 0.440·53-s − 1.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - 841T \) |
| good | 3 | \( 1 + 29.3T + 243T^{2} \) |
| 5 | \( 1 - 64.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 138.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 557.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 41.1T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.64e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 258.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.82e3T + 6.43e6T^{2} \) |
| 31 | \( 1 - 5.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.32e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.89e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.58e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.50e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.01e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.95e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.12e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.39e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.02e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.55e4T + 8.58e9T^{2} \) |
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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
−10.28834224368758972605891792333, −9.077573011752028566677195086607, −7.74569562427429563329859434019, −6.70645995883324458025881405647, −5.90634954793929794583660752609, −5.04686436234686137485724007279, −4.63354387697478698510248844599, −2.32247950444102598920501786400, −1.26350802714068901200933363607, 0,
1.26350802714068901200933363607, 2.32247950444102598920501786400, 4.63354387697478698510248844599, 5.04686436234686137485724007279, 5.90634954793929794583660752609, 6.70645995883324458025881405647, 7.74569562427429563329859434019, 9.077573011752028566677195086607, 10.28834224368758972605891792333
