| L(s) = 1 | + (−0.730 − 1.86i)2-s + (−0.524 − 6.99i)3-s + (−2.93 + 2.72i)4-s + (−5.50 + 3.75i)5-s + (−12.6 + 6.08i)6-s + (−17.1 + 6.93i)7-s + (7.20 + 3.47i)8-s + (−22.0 + 3.31i)9-s + (11.0 + 7.50i)10-s + (−9.03 − 1.36i)11-s + (20.5 + 19.0i)12-s + (−4.48 + 5.62i)13-s + (25.4 + 26.8i)14-s + (29.1 + 36.5i)15-s + (1.19 − 15.9i)16-s + (35.8 − 11.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.258 − 0.658i)2-s + (−0.100 − 1.34i)3-s + (−0.366 + 0.340i)4-s + (−0.492 + 0.335i)5-s + (−0.860 + 0.414i)6-s + (−0.927 + 0.374i)7-s + (0.318 + 0.153i)8-s + (−0.814 + 0.122i)9-s + (0.348 + 0.237i)10-s + (−0.247 − 0.0373i)11-s + (0.495 + 0.459i)12-s + (−0.0957 + 0.120i)13-s + (0.486 + 0.513i)14-s + (0.501 + 0.629i)15-s + (0.0186 − 0.249i)16-s + (0.512 − 0.157i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0849348 + 0.111164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0849348 + 0.111164i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.730 + 1.86i)T \) |
| 7 | \( 1 + (17.1 - 6.93i)T \) |
| good | 3 | \( 1 + (0.524 + 6.99i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (5.50 - 3.75i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (9.03 + 1.36i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (4.48 - 5.62i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-35.8 + 11.0i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (27.4 - 47.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (154. + 47.6i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (39.3 - 172. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (111. + 192. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (268. + 249. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-143. - 69.2i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (0.691 - 0.333i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (83.5 + 212. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (107. - 100. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-14.8 - 10.0i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (251. + 233. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (521. + 903. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-92.9 - 407. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (61.9 - 157. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (144. - 250. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-729. - 914. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-789. + 118. i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−12.48461257368988798982223153816, −11.96235942218684977844770236920, −10.68356490662863885064611313970, −9.433957998989024501534173416716, −8.054720015613132288167557895667, −7.14183521634857312479275880636, −5.88626120710171944352209801536, −3.57824659598442908048024523566, −2.04032492051316504346670457111, −0.084897145553494445057338904152,
3.62061260320855147844157397570, 4.70114809192083085652298434823, 6.06190447107579188895464958434, 7.57534958102341208629052705820, 8.846755023445726047859875676365, 9.896653224517400497683396537195, 10.50164149434689709178553663365, 12.00440840090377092829960109730, 13.29787508353383225454666345275, 14.48269974641876530122444808581
