| L(s) = 1 | + 4·2-s + 16·4-s − 81·5-s − 247·7-s + 64·8-s − 324·10-s + 465·11-s − 694·13-s − 988·14-s + 256·16-s − 543·17-s + 361·19-s − 1.29e3·20-s + 1.86e3·22-s + 2.72e3·23-s + 3.43e3·25-s − 2.77e3·26-s − 3.95e3·28-s − 342·29-s − 9.44e3·31-s + 1.02e3·32-s − 2.17e3·34-s + 2.00e4·35-s + 1.30e4·37-s + 1.44e3·38-s − 5.18e3·40-s + 1.62e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.44·5-s − 1.90·7-s + 0.353·8-s − 1.02·10-s + 1.15·11-s − 1.13·13-s − 1.34·14-s + 1/4·16-s − 0.455·17-s + 0.229·19-s − 0.724·20-s + 0.819·22-s + 1.07·23-s + 1.09·25-s − 0.805·26-s − 0.952·28-s − 0.0755·29-s − 1.76·31-s + 0.176·32-s − 0.322·34-s + 2.76·35-s + 1.57·37-s + 0.162·38-s − 0.512·40-s + 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.448077460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.448077460\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 81 T + p^{5} T^{2} \) |
| 7 | \( 1 + 247 T + p^{5} T^{2} \) |
| 11 | \( 1 - 465 T + p^{5} T^{2} \) |
| 13 | \( 1 + 694 T + p^{5} T^{2} \) |
| 17 | \( 1 + 543 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2724 T + p^{5} T^{2} \) |
| 29 | \( 1 + 342 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9442 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13088 T + p^{5} T^{2} \) |
| 41 | \( 1 - 16272 T + p^{5} T^{2} \) |
| 43 | \( 1 + 391 T + p^{5} T^{2} \) |
| 47 | \( 1 - 8523 T + p^{5} T^{2} \) |
| 53 | \( 1 - 10110 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27144 T + p^{5} T^{2} \) |
| 61 | \( 1 + 48829 T + p^{5} T^{2} \) |
| 67 | \( 1 - 55448 T + p^{5} T^{2} \) |
| 71 | \( 1 + 43212 T + p^{5} T^{2} \) |
| 73 | \( 1 - 37685 T + p^{5} T^{2} \) |
| 79 | \( 1 + 78016 T + p^{5} T^{2} \) |
| 83 | \( 1 + 83892 T + p^{5} T^{2} \) |
| 89 | \( 1 + 25530 T + p^{5} T^{2} \) |
| 97 | \( 1 + 76378 T + p^{5} T^{2} \) |
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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
−10.99884925775512651991669944104, −9.690083045827659150398795084560, −8.983433933222850345126771409338, −7.38765935301421300148359759683, −6.97556996845918473883555130934, −5.87752122676405345833776140464, −4.37416385393654349305736309030, −3.65480245572807966689560434822, −2.73798406649462804610211974490, −0.56788885782657052305213356753,
0.56788885782657052305213356753, 2.73798406649462804610211974490, 3.65480245572807966689560434822, 4.37416385393654349305736309030, 5.87752122676405345833776140464, 6.97556996845918473883555130934, 7.38765935301421300148359759683, 8.983433933222850345126771409338, 9.690083045827659150398795084560, 10.99884925775512651991669944104
