| L(s) = 1 | + (−3.43 − 3.43i)5-s − 8.14·7-s + (−1.92 + 1.92i)11-s + (9.16 + 9.16i)13-s − 28.6i·17-s + (−39.9 + 39.9i)19-s + 80.3i·23-s − 101. i·25-s + (113. − 113. i)29-s + 306. i·31-s + (27.9 + 27.9i)35-s + (47.8 − 47.8i)37-s + 349.·41-s + (164. + 164. i)43-s + 40.5·47-s + ⋯ |
| L(s) = 1 | + (−0.307 − 0.307i)5-s − 0.439·7-s + (−0.0526 + 0.0526i)11-s + (0.195 + 0.195i)13-s − 0.409i·17-s + (−0.481 + 0.481i)19-s + 0.728i·23-s − 0.811i·25-s + (0.729 − 0.729i)29-s + 1.77i·31-s + (0.135 + 0.135i)35-s + (0.212 − 0.212i)37-s + 1.33·41-s + (0.583 + 0.583i)43-s + 0.125·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.491167692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491167692\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.43 + 3.43i)T + 125iT^{2} \) |
| 7 | \( 1 + 8.14T + 343T^{2} \) |
| 11 | \( 1 + (1.92 - 1.92i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (-9.16 - 9.16i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + 28.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (39.9 - 39.9i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 80.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-113. + 113. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 306. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-47.8 + 47.8i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 349.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-164. - 164. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 - 40.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-454. - 454. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-245. + 245. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-186. - 186. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (-118. + 118. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 414. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 431. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-739. - 739. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 942.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 983.T + 9.12e5T^{2} \) |
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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
−10.37097308165345428114999448459, −9.528368447932968462417299484559, −8.644827035319493396331689565678, −7.80834430813762013096954952870, −6.78320569408991919692932675415, −5.88316392246362956953095737257, −4.72130057507920222266769661666, −3.75811566724025877636598031394, −2.51195087579732672346649468960, −0.937973931979613030708350217117,
0.57459262653604801297584539155, 2.31265121016701440879793439718, 3.45881576763164095608987091037, 4.47227032681117945398374143528, 5.74455118491868865751291333166, 6.61586060819968397282176364976, 7.52554209747878301339470356967, 8.476481481705604878551243844499, 9.343794989009837165850455396410, 10.32424206989879376173872767315
