| L(s) = 1 | − 4.53·2-s − 15.7·3-s − 11.4·4-s − 25·5-s + 71.2·6-s + 49·7-s + 196.·8-s + 4.27·9-s + 113.·10-s + 126.·11-s + 180.·12-s + 833.·13-s − 222.·14-s + 393.·15-s − 526.·16-s + 157.·17-s − 19.3·18-s − 1.00e3·19-s + 286.·20-s − 770.·21-s − 572.·22-s + 73.5·23-s − 3.09e3·24-s + 625·25-s − 3.77e3·26-s + 3.75e3·27-s − 561.·28-s + ⋯ |
| L(s) = 1 | − 0.801·2-s − 1.00·3-s − 0.357·4-s − 0.447·5-s + 0.808·6-s + 0.377·7-s + 1.08·8-s + 0.0175·9-s + 0.358·10-s + 0.314·11-s + 0.360·12-s + 1.36·13-s − 0.302·14-s + 0.451·15-s − 0.514·16-s + 0.132·17-s − 0.0140·18-s − 0.638·19-s + 0.160·20-s − 0.381·21-s − 0.252·22-s + 0.0289·23-s − 1.09·24-s + 0.200·25-s − 1.09·26-s + 0.991·27-s − 0.135·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5729627883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5729627883\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 - 49T \) |
| good | 2 | \( 1 + 4.53T + 32T^{2} \) |
| 3 | \( 1 + 15.7T + 243T^{2} \) |
| 11 | \( 1 - 126.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 833.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 157.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 73.5T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.63e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.36e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.67e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.79e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 195.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.60e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.58e5T + 8.58e9T^{2} \) |
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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
−15.98927087136655674128275839102, −14.36230972901366394660993735317, −12.94124877014136021941133912820, −11.43076421324612234543409220125, −10.70076278139340885042628623751, −9.065478249107606372137347148100, −7.88318914689567097397983394317, −6.05743112285759524621893863763, −4.34848426983384825448730978396, −0.852884046253631437727049846964,
0.852884046253631437727049846964, 4.34848426983384825448730978396, 6.05743112285759524621893863763, 7.88318914689567097397983394317, 9.065478249107606372137347148100, 10.70076278139340885042628623751, 11.43076421324612234543409220125, 12.94124877014136021941133912820, 14.36230972901366394660993735317, 15.98927087136655674128275839102
