L(s) = 1 | + 2i·2-s + 7.11i·3-s − 4·4-s + (−6.02 + 9.41i)5-s − 14.2·6-s − 8i·8-s − 23.5·9-s + (−18.8 − 12.0i)10-s − 65.2·11-s − 28.4i·12-s + 55.0i·13-s + (−66.9 − 42.8i)15-s + 16·16-s − 116. i·17-s − 47.1i·18-s + 98.8·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.36i·3-s − 0.5·4-s + (−0.539 + 0.842i)5-s − 0.967·6-s − 0.353i·8-s − 0.872·9-s + (−0.595 − 0.381i)10-s − 1.78·11-s − 0.684i·12-s + 1.17i·13-s + (−1.15 − 0.737i)15-s + 0.250·16-s − 1.66i·17-s − 0.616i·18-s + 1.19·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2253088330\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2253088330\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 + (6.02 - 9.41i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 7.11iT - 27T^{2} \) |
| 11 | \( 1 + 65.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 116. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 98.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 47.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 107. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 280.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 236. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 85.8iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 473. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 548.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 693.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 310. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 331.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 277. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 645.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 315. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 829.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 63.7iT - 9.12e5T^{2} \) |
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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
−11.21996167160632641896645887123, −10.23990048050537408007510276428, −9.699930954362177659643977625859, −8.758838529653736162445701850203, −7.53672832004011920623473646388, −7.04193881647108904809807249245, −5.45927759501551466697454577541, −4.88497873857529119385254500460, −3.74802928523541020156249171187, −2.78415027232850430471751013112,
0.080451029830261702255471840556, 1.09186662904159029448950018037, 2.30386095606353051502059171435, 3.51806933257506551417198001839, 5.03042053881121873840719811785, 5.78768524580364271446944287255, 7.28805069889907100066046763950, 8.143933455576054940971627065962, 8.377275713885601298018699986289, 9.944939169149272814288305418324