L(s) = 1 | + (0.0399 − 0.0399i)3-s + (−1.97 + 1.97i)5-s + (−4.30 − 4.30i)7-s + 26.9i·9-s + (36.5 + 36.5i)11-s + 19.2·13-s + 0.158i·15-s + (−57.9 + 39.4i)17-s + 25.0i·19-s − 0.343·21-s + (105. + 105. i)23-s + 117. i·25-s + (2.15 + 2.15i)27-s + (−14.6 + 14.6i)29-s + (161. − 161. i)31-s + ⋯ |
L(s) = 1 | + (0.00769 − 0.00769i)3-s + (−0.177 + 0.177i)5-s + (−0.232 − 0.232i)7-s + 0.999i·9-s + (1.00 + 1.00i)11-s + 0.410·13-s + 0.00272i·15-s + (−0.826 + 0.563i)17-s + 0.301i·19-s − 0.00357·21-s + (0.959 + 0.959i)23-s + 0.937i·25-s + (0.0153 + 0.0153i)27-s + (−0.0940 + 0.0940i)29-s + (0.935 − 0.935i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.17155 + 0.855293i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17155 + 0.855293i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (57.9 - 39.4i)T \) |
good | 3 | \( 1 + (-0.0399 + 0.0399i)T - 27iT^{2} \) |
| 5 | \( 1 + (1.97 - 1.97i)T - 125iT^{2} \) |
| 7 | \( 1 + (4.30 + 4.30i)T + 343iT^{2} \) |
| 11 | \( 1 + (-36.5 - 36.5i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 - 19.2T + 2.19e3T^{2} \) |
| 19 | \( 1 - 25.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-105. - 105. i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (14.6 - 14.6i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + (-161. + 161. i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (249. - 249. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (155. + 155. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + 51.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 261.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 102. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 694. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + (154. + 154. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-382. + 382. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (190. - 190. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (22.3 + 22.3i)T + 4.93e5iT^{2} \) |
| 83 | \( 1 - 669. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-218. + 218. i)T - 9.12e5iT^{2} \) |
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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
−13.08513422564920765331674126103, −11.84996943156947127090609586322, −10.93581081518437395667566717467, −9.898282950928659386812571311562, −8.768323735858833154097003307019, −7.50297822871348641591470519448, −6.55178737365624546354335251814, −4.97967712506638171825246227642, −3.68090103046209387060104221785, −1.78106861871324476516998552857,
0.77693773535520826600792917448, 3.06251982269827302078760352207, 4.39401589859771642932903003560, 6.08191947731073583829921637519, 6.88780254912224468916930261196, 8.705620989518892862628241647748, 9.057808530126462458685843325843, 10.57486226860323044301864974617, 11.65150103541620850398835864578, 12.39361635237024285996573544170