| L(s) = 1 | + (−2.78 − 3.20i)2-s + (0.620 + 0.398i)3-s + (−1.42 + 9.93i)4-s + (2.07 − 4.54i)5-s + (−0.446 − 3.10i)6-s + (−1.32 − 0.388i)7-s + (7.27 − 4.67i)8-s + (−10.9 − 24.0i)9-s + (−20.3 + 5.98i)10-s + (−14.8 + 17.1i)11-s + (−4.85 + 5.59i)12-s + (−74.9 + 22.0i)13-s + (2.43 + 5.32i)14-s + (3.10 − 1.99i)15-s + (41.7 + 12.2i)16-s + (11.2 + 77.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.983 − 1.13i)2-s + (0.119 + 0.0767i)3-s + (−0.178 + 1.24i)4-s + (0.185 − 0.406i)5-s + (−0.0303 − 0.211i)6-s + (−0.0713 − 0.0209i)7-s + (0.321 − 0.206i)8-s + (−0.407 − 0.891i)9-s + (−0.644 + 0.189i)10-s + (−0.407 + 0.470i)11-s + (−0.116 + 0.134i)12-s + (−1.60 + 0.469i)13-s + (0.0464 + 0.101i)14-s + (0.0534 − 0.0343i)15-s + (0.652 + 0.191i)16-s + (0.159 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.99528\times10^{-5} + 4.49470\times10^{-5}i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.99528\times10^{-5} + 4.49470\times10^{-5}i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.07 + 4.54i)T \) |
| 23 | \( 1 + (34.9 - 104. i)T \) |
| good | 2 | \( 1 + (2.78 + 3.20i)T + (-1.13 + 7.91i)T^{2} \) |
| 3 | \( 1 + (-0.620 - 0.398i)T + (11.2 + 24.5i)T^{2} \) |
| 7 | \( 1 + (1.32 + 0.388i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (14.8 - 17.1i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (74.9 - 22.0i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-11.2 - 77.9i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-15.1 + 105. i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-37.5 - 261. i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (169. - 108. i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-55.8 - 122. i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-154. + 337. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (376. + 242. i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 + 376.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (236. + 69.5i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (93.6 - 27.5i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-650. + 417. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (147. + 169. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-178. - 205. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-56.5 + 393. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (17.1 - 5.03i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-362. - 793. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (535. + 344. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-241. + 527. i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−12.21091698475040698813473214711, −11.20226840065720622148381285895, −10.00706127796421050251651776530, −9.367531168822833278132917647139, −8.457681728309608362451593618917, −7.01346605814738011434599463480, −5.16544046597213070894008942469, −3.32203496942244072392678008414, −1.84792415876142234700126695734, −0.00004112996519286253218648726,
2.67621712119839734428916389165, 5.16398845258513166350917429852, 6.27325056959775251881272233185, 7.68337230807013327637699582075, 8.007069620910015031203990717996, 9.570564005272895045417092233151, 10.20083432807247326583734543391, 11.60498366018056294564443952958, 12.99382304853253867454427088533, 14.33138712292448877282841328185
