| L(s) = 1 | + (1.91 − 0.563i)2-s + (−4.61 − 5.32i)3-s + (3.36 − 2.16i)4-s + (−0.711 − 4.94i)5-s + (−11.8 − 7.61i)6-s + (10.2 − 22.4i)7-s + (5.23 − 6.04i)8-s + (−3.22 + 22.4i)9-s + (−4.15 − 9.09i)10-s + (−2.47 − 0.728i)11-s + (−27.0 − 7.94i)12-s + (−2.22 − 4.86i)13-s + (7.02 − 48.8i)14-s + (−23.0 + 26.6i)15-s + (6.64 − 14.5i)16-s + (−16.7 − 10.7i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (−0.888 − 1.02i)3-s + (0.420 − 0.270i)4-s + (−0.0636 − 0.442i)5-s + (−0.806 − 0.518i)6-s + (0.553 − 1.21i)7-s + (0.231 − 0.267i)8-s + (−0.119 + 0.830i)9-s + (−0.131 − 0.287i)10-s + (−0.0679 − 0.0199i)11-s + (−0.650 − 0.191i)12-s + (−0.0473 − 0.103i)13-s + (0.134 − 0.932i)14-s + (−0.397 + 0.458i)15-s + (0.103 − 0.227i)16-s + (−0.239 − 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0217i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0175581 - 1.61747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0175581 - 1.61747i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.91 + 0.563i)T \) |
| 5 | \( 1 + (0.711 + 4.94i)T \) |
| 23 | \( 1 + (-63.7 - 90.0i)T \) |
| good | 3 | \( 1 + (4.61 + 5.32i)T + (-3.84 + 26.7i)T^{2} \) |
| 7 | \( 1 + (-10.2 + 22.4i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (2.47 + 0.728i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (2.22 + 4.86i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (16.7 + 10.7i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (22.8 - 14.6i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (187. + 120. i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (94.2 - 108. i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (-54.9 + 382. i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-18.6 - 129. i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-214. - 247. i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 - 45.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (80.2 - 175. i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (168. + 369. i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (-267. + 308. i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (-596. + 175. i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-678. + 199. i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (-500. + 321. i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (324. + 710. i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-51.4 + 357. i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (301. + 347. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (69.3 + 482. i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
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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
−11.22857426316715933967618663306, −10.97363894710799268972246684350, −9.477768423594852665308212414702, −7.78436620467216233712427201639, −7.17859460901244581730000456052, −6.03566769522884923498397897206, −5.02744843498941800097439347600, −3.84411639870929380225185214319, −1.75567423106403427619715029567, −0.59085398758031610046167834792,
2.39837763243057657028547664969, 3.96435199452779646489739377585, 5.07711128678803015131175459438, 5.71811304846886457710979631673, 6.86376525276157895729776326706, 8.332851746263918872865919375108, 9.444544391975687615379900264244, 10.68583979276013691773011052622, 11.26098454465012264214393015333, 12.04656745209078345254749836620
