| L(s) = 1 | + (−4.17 + 3.03i)2-s + (1.40 + 4.31i)3-s + (5.75 − 17.7i)4-s + (6.98 + 5.07i)5-s + (−18.9 − 13.7i)6-s + (0.513 − 1.58i)7-s + (16.9 + 52.1i)8-s + (5.15 − 3.74i)9-s − 44.5·10-s + (−26.2 − 25.3i)11-s + 84.5·12-s + (23.1 − 16.8i)13-s + (2.65 + 8.15i)14-s + (−12.1 + 37.2i)15-s + (−108. − 78.6i)16-s + (−6.36 − 4.62i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 + 1.07i)2-s + (0.270 + 0.831i)3-s + (0.719 − 2.21i)4-s + (0.624 + 0.453i)5-s + (−1.28 − 0.937i)6-s + (0.0277 − 0.0853i)7-s + (0.748 + 2.30i)8-s + (0.191 − 0.138i)9-s − 1.40·10-s + (−0.719 − 0.694i)11-s + 2.03·12-s + (0.494 − 0.358i)13-s + (0.0506 + 0.155i)14-s + (−0.208 + 0.642i)15-s + (−1.69 − 1.22i)16-s + (−0.0907 − 0.0659i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.365704 + 0.425396i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.365704 + 0.425396i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (26.2 + 25.3i)T \) |
| good | 2 | \( 1 + (4.17 - 3.03i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (-1.40 - 4.31i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-6.98 - 5.07i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-0.513 + 1.58i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-23.1 + 16.8i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (6.36 + 4.62i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-14.8 - 45.6i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-74.9 + 230. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (134. - 97.5i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (21.2 - 65.4i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-95.0 - 292. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 52.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + (16.8 + 51.8i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (202. - 147. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-117. + 360. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (565. + 410. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 278.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-125. - 90.9i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (293. - 904. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (1.79 - 1.30i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-614. - 446. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (598. - 434. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−20.27590325450522842267171376876, −18.64767109659193475910909460395, −17.82169967982785171907834871499, −16.29794820778511096742650159002, −15.48333753302076647203353175001, −14.09504751333778429045235428592, −10.57941091108065317410326323355, −9.701436612108996873812731909927, −8.118902827648544348987814476459, −6.08933254733679359825471308870,
1.90141468123905850216987321031, 7.49525610904869238059665527215, 9.002443681269460045016356620077, 10.45476061694864452690370398450, 12.24860265273379933892510847996, 13.36083439366785730285016658716, 16.23137863381728838204098773632, 17.76671785230457100154887658197, 18.38417398666348919051138129766, 19.64656545974041844833800549140
