| L(s) = 1 | + (−1.95 − 15.4i)3-s − 34.2i·5-s + 49i·7-s + (−235. + 60.4i)9-s + 609.·11-s − 164.·13-s + (−530. + 66.9i)15-s + 866. i·17-s + 535. i·19-s + (757. − 95.7i)21-s − 4.04e3·23-s + 1.95e3·25-s + (1.39e3 + 3.52e3i)27-s + 7.21e3i·29-s + 4.42e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.125 − 0.992i)3-s − 0.613i·5-s + 0.377i·7-s + (−0.968 + 0.248i)9-s + 1.52·11-s − 0.270·13-s + (−0.608 + 0.0768i)15-s + 0.727i·17-s + 0.340i·19-s + (0.374 − 0.0473i)21-s − 1.59·23-s + 0.624·25-s + (0.368 + 0.929i)27-s + 1.59i·29-s + 0.826i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.703289753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.703289753\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.95 + 15.4i)T \) |
| 7 | \( 1 - 49iT \) |
| good | 5 | \( 1 + 34.2iT - 3.12e3T^{2} \) |
| 11 | \( 1 - 609.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 164.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 866. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 535. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 4.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.21e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.42e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 7.28e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.00e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.86e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 931.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.02e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.80e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.50e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.28e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.75e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 8.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.28e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 8.52e4T + 8.58e9T^{2} \) |
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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
−10.97102105916985390420269924062, −9.626180185552961390215759954707, −8.706939436311819346489591375530, −8.015489154612460425073244830815, −6.74640306266322045721188634917, −6.08046267227325214504471225218, −4.88582311216330506539084813356, −3.50151665184726775261830743137, −1.91310319571199556615680073581, −1.05331848844718047677230010862,
0.52353228746052352777323877988, 2.43322879358614652920068444994, 3.75553696199986231608708484052, 4.43536923402760463454329188991, 5.85416893218243382136028843336, 6.70969402092171570607702645750, 7.893611703405356385625952371329, 9.166879790146426358836639267085, 9.749266030625740970229479929010, 10.66318875655345157327221074459
