| L(s) = 1 | + 23·2-s + 273·4-s + 625·5-s − 1.28e3·7-s + 391·8-s + 6.56e3·9-s + 1.43e4·10-s + 1.46e4·11-s + 3.08e4·13-s − 2.94e4·14-s − 6.08e4·16-s + 5.43e3·17-s + 1.50e5·18-s + 1.70e5·20-s + 3.36e5·22-s + 3.90e5·25-s + 7.10e5·26-s − 3.49e5·28-s + 7.06e5·31-s − 1.50e6·32-s + 1.25e5·34-s − 8.01e5·35-s + 1.79e6·36-s + 2.44e5·40-s − 5.56e6·43-s + 3.99e6·44-s + 4.10e6·45-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.06·4-s + 5-s − 0.533·7-s + 0.0954·8-s + 9-s + 1.43·10-s + 11-s + 1.08·13-s − 0.767·14-s − 0.929·16-s + 0.0651·17-s + 1.43·18-s + 1.06·20-s + 1.43·22-s + 25-s + 1.55·26-s − 0.569·28-s + 0.765·31-s − 1.43·32-s + 0.0935·34-s − 0.533·35-s + 1.06·36-s + 0.0954·40-s − 1.62·43-s + 1.06·44-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(5.097290471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.097290471\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{4} T \) |
| 11 | \( 1 - p^{4} T \) |
| good | 2 | \( 1 - 23 T + p^{8} T^{2} \) |
| 3 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( 1 + 1282 T + p^{8} T^{2} \) |
| 13 | \( 1 - 30878 T + p^{8} T^{2} \) |
| 17 | \( 1 - 5438 T + p^{8} T^{2} \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 - 706562 T + p^{8} T^{2} \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 + 5566882 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( 1 + 12387358 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 71 | \( 1 + 34839358 T + p^{8} T^{2} \) |
| 73 | \( 1 + 56370562 T + p^{8} T^{2} \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( 1 + 90097762 T + p^{8} T^{2} \) |
| 89 | \( 1 - 125357762 T + p^{8} T^{2} \) |
| 97 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
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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
−13.46805599807113916126373097836, −12.88879745906531170517663779001, −11.69010768128392636120305985150, −10.18084731721059754306863400926, −9.011301176828185366444771527483, −6.73932674407237765751047803817, −6.00765872990047600348997721207, −4.53036697988747383598357700751, −3.28008397144105355325079195004, −1.52909423904019615058239919400,
1.52909423904019615058239919400, 3.28008397144105355325079195004, 4.53036697988747383598357700751, 6.00765872990047600348997721207, 6.73932674407237765751047803817, 9.011301176828185366444771527483, 10.18084731721059754306863400926, 11.69010768128392636120305985150, 12.88879745906531170517663779001, 13.46805599807113916126373097836
