| L(s) = 1 | + (−1.53 − 1.28i)2-s + (−2.81 + 1.02i)3-s + (0.694 + 3.93i)4-s + (0.424 − 2.40i)5-s + (5.63 + 2.05i)6-s + (−6.23 − 10.7i)7-s + (4.00 − 6.92i)8-s + (6.89 − 5.78i)9-s + (−3.74 + 3.14i)10-s + (−32.0 + 55.4i)11-s + (−6 − 10.3i)12-s + (62.1 + 22.6i)13-s + (−4.32 + 24.5i)14-s + (1.27 + 7.22i)15-s + (−15.0 + 5.47i)16-s + (60.5 + 50.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.542 + 0.197i)3-s + (0.0868 + 0.492i)4-s + (0.0380 − 0.215i)5-s + (0.383 + 0.139i)6-s + (−0.336 − 0.582i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.118 + 0.0994i)10-s + (−0.877 + 1.51i)11-s + (−0.144 − 0.249i)12-s + (1.32 + 0.482i)13-s + (−0.0826 + 0.468i)14-s + (0.0219 + 0.124i)15-s + (−0.234 + 0.0855i)16-s + (0.864 + 0.725i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.832664 + 0.291160i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.832664 + 0.291160i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.53 + 1.28i)T \) |
| 3 | \( 1 + (2.81 - 1.02i)T \) |
| 19 | \( 1 + (-76.6 - 31.4i)T \) |
| good | 5 | \( 1 + (-0.424 + 2.40i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (6.23 + 10.7i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (32.0 - 55.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-62.1 - 22.6i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-60.5 - 50.8i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (13.7 + 78.1i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (154. - 129. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-134. - 233. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-161. + 58.7i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-7.38 + 41.9i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (223. - 187. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-75.0 - 425. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (598. + 501. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (48.2 + 273. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (29.2 - 24.5i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-77.4 + 439. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-1.08e3 + 394. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (759. - 276. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-626. - 1.08e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (478. + 174. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-690. - 579. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−12.80908521920205890285717546951, −12.25482837620235987874698998052, −10.81404291000623199903673888933, −10.26775131249368226234432834187, −9.168106157355951650986301938717, −7.79501916437318690205719635338, −6.66047432365271974584078434880, −5.02740061621885242385418739979, −3.57162868162113201527155161756, −1.38522910308526274074505735416,
0.69098819313909018443061003294, 3.09339935221098871666291291537, 5.49981037933657035804543900717, 6.05443583486612144145070640535, 7.55988807325075284130053340537, 8.548982952804063996067673454573, 9.781401562287509874265069118552, 10.94358519078683511804955605504, 11.66202231675628516486666750579, 13.18684892384497341898340791027
