| L(s) = 1 | − 8.69i·3-s + 32.3·5-s + 39.7·7-s + 5.41·9-s + 105. i·11-s + 26.9i·13-s − 280. i·15-s + 100. i·17-s + 382.·19-s − 345. i·21-s − 717. i·23-s + 418.·25-s − 751. i·27-s − 448. i·29-s + (−658. − 699. i)31-s + ⋯ |
| L(s) = 1 | − 0.966i·3-s + 1.29·5-s + 0.810·7-s + 0.0668·9-s + 0.873i·11-s + 0.159i·13-s − 1.24i·15-s + 0.346i·17-s + 1.05·19-s − 0.782i·21-s − 1.35i·23-s + 0.670·25-s − 1.03i·27-s − 0.532i·29-s + (−0.685 − 0.728i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.27904 - 0.984940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.27904 - 0.984940i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (658. + 699. i)T \) |
| good | 3 | \( 1 + 8.69iT - 81T^{2} \) |
| 5 | \( 1 - 32.3T + 625T^{2} \) |
| 7 | \( 1 - 39.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 105. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 26.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 100. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 382.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 717. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 448. iT - 7.07e5T^{2} \) |
| 37 | \( 1 - 2.19e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 15.3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.30e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 250.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 3.76e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 460.T + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.52e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.82e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.57e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 2.84e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 8.01e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 5.38e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.70e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.70e3T + 8.85e7T^{2} \) |
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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
−12.69559059251912020581348873994, −11.76029494334303249166413011473, −10.37275869273175411164651278656, −9.482566646701939522627152249374, −8.099643559912621706511364961809, −7.03951599618424347776309885830, −5.98480009451854525440828300739, −4.66319691628919329111240811007, −2.27860280041002827827254035919, −1.35220660067368787571104538471,
1.54157197554776294718236694487, 3.37770274580925805380225089120, 5.01090801988025431440968249499, 5.74518415964194212811695789617, 7.42698063777453485765861830792, 8.950600805210854688686493158237, 9.680043584927500536460564304582, 10.64001359710881269139325434486, 11.51471351171343306391257671806, 13.06201376552529667212091391333
