L(s) = 1 | + (−23.4 − 50.7i)5-s − 10.2i·7-s + 596.·11-s + 420. i·13-s − 974. i·17-s − 380.·19-s − 3.54e3i·23-s + (−2.02e3 + 2.37e3i)25-s + 5.44e3·29-s + 3.62e3·31-s + (−520. + 240. i)35-s + 1.75e3i·37-s − 263.·41-s + 1.44e4i·43-s + 2.34e4i·47-s + ⋯ |
L(s) = 1 | + (−0.419 − 0.907i)5-s − 0.0791i·7-s + 1.48·11-s + 0.690i·13-s − 0.817i·17-s − 0.241·19-s − 1.39i·23-s + (−0.648 + 0.760i)25-s + 1.20·29-s + 0.677·31-s + (−0.0718 + 0.0331i)35-s + 0.210i·37-s − 0.0245·41-s + 1.18i·43-s + 1.54i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.180456982\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.180456982\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (23.4 + 50.7i)T \) |
good | 7 | \( 1 + 10.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 596.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 420. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 974. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 380.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.54e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.75e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 263.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.44e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.34e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.34e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.16e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 8.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.51e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.29e4iT - 8.58e9T^{2} \) |
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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
−9.208590023973272690019510471023, −8.845130782792315103017036181327, −7.84630503731692950983129998300, −6.79839216505260951577702818652, −6.06472744214054026746283679133, −4.53688005425024073516702846052, −4.36288203381711057270607237504, −2.90515951120082088713759945014, −1.45879027932858330539191127212, −0.60357518105335594277876384796,
0.888419825970146979649165215715, 2.17835924943414403065866381828, 3.45599059610337497261531802066, 4.01860627971141916130710081741, 5.44809373050384889549322766315, 6.44354446838590481680849960185, 7.05866697624885833369668296762, 8.098679467866470437476765828375, 8.861173133454724095286647330851, 9.993287014828751936402849190835