L(s) = 1 | − 2·2-s + (2.99 − 4.24i)3-s + 4·4-s + 19.3i·5-s + (−5.98 + 8.49i)6-s + 6.85·7-s − 8·8-s + (−9.10 − 25.4i)9-s − 38.6i·10-s − 4.18·11-s + (11.9 − 16.9i)12-s + 24.0i·13-s − 13.7·14-s + (82.0 + 57.7i)15-s + 16·16-s + 81.9i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.575 − 0.817i)3-s + 0.5·4-s + 1.72i·5-s + (−0.407 + 0.578i)6-s + 0.370·7-s − 0.353·8-s + (−0.337 − 0.941i)9-s − 1.22i·10-s − 0.114·11-s + (0.287 − 0.408i)12-s + 0.513i·13-s − 0.261·14-s + (1.41 + 0.994i)15-s + 0.250·16-s + 1.16i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0408 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0408 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.253697711\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.253697711\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + (-2.99 + 4.24i)T \) |
| 59 | \( 1 + (-359. + 275. i)T \) |
good | 5 | \( 1 - 19.3iT - 125T^{2} \) |
| 7 | \( 1 - 6.85T + 343T^{2} \) |
| 11 | \( 1 + 4.18T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 81.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 216.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 136. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 342. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 346. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 280. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 353. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 88.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 89.3iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 198. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 856. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 384. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 586. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 417.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 329.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 49.9iT - 9.12e5T^{2} \) |
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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
−11.42352101401501368453978493581, −10.18484809643485780612469267788, −9.572805374141250426308448491670, −8.075079299386834591711882733217, −7.73625117171162942633220560590, −6.64308572488989570352545491853, −6.04873308316580048319878221764, −3.72018079644094806969368308834, −2.66816855649253083141515286276, −1.61179767229924186699662177754,
0.50402295662734037370960365797, 2.00379208946859965956701723867, 3.63588987143886407326887598420, 4.92961745004767585417916959322, 5.53713884312923654831402783583, 7.55914598588295856645773643319, 8.198769151854884273440840207941, 9.042788727043410355526107717740, 9.615370631462079842132867037985, 10.49100261070667795217966052299