| 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
−12.04656745209078345254749836620, −11.26098454465012264214393015333, −10.68583979276013691773011052622, −9.444544391975687615379900264244, −8.332851746263918872865919375108, −6.86376525276157895729776326706, −5.71811304846886457710979631673, −5.07711128678803015131175459438, −3.96435199452779646489739377585, −2.39837763243057657028547664969,
0.59085398758031610046167834792, 1.75567423106403427619715029567, 3.84411639870929380225185214319, 5.02744843498941800097439347600, 6.03566769522884923498397897206, 7.17859460901244581730000456052, 7.78436620467216233712427201639, 9.477768423594852665308212414702, 10.97363894710799268972246684350, 11.22857426316715933967618663306
