| L(s) = 1 | + 4i·2-s − 9i·3-s − 16·4-s + 36·6-s − 233i·7-s − 64i·8-s − 81·9-s − 498·11-s + 144i·12-s + 809i·13-s + 932·14-s + 256·16-s + 1.00e3i·17-s − 324i·18-s + 1.70e3·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.79i·7-s − 0.353i·8-s − 0.333·9-s − 1.24·11-s + 0.288i·12-s + 1.32i·13-s + 1.27·14-s + 0.250·16-s + 0.840i·17-s − 0.235i·18-s + 1.08·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3438557011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3438557011\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 + 9iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 233iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 498T + 1.61e5T^{2} \) |
| 13 | \( 1 - 809iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.00e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.55e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 7.83e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 977T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.82e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 8.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.94e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.41e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.76e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.52e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.74e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.54e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 646iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.15e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 7.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.45e4iT - 8.58e9T^{2} \) |
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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
−12.96489111976633540213824815071, −11.48370053579559548376354925762, −10.46643105410247251023618394375, −9.391195532199598322769341869630, −7.84921768110276961734355967325, −7.39896658181839638152734815022, −6.31996262180071977135156809391, −4.87694655422163909207611297570, −3.60329250002897021904156373205, −1.43400060347180561047025884821,
0.11622926243884495957773093785, 2.37625841979734407820014323027, 3.21359911207452515415251330428, 5.19191281937103878960077941928, 5.56630319615986405776931572877, 7.77301808476763418780870853495, 8.814147246930874808940940904809, 9.685607023045799140914612688712, 10.66862813160359640644522560363, 11.67805108466597466412121554244
