| L(s) = 1 | + (−0.5 + 0.866i)2-s + (2 − 3.46i)3-s + (3.5 + 6.06i)4-s + (0.982 + 0.567i)5-s + (1.99 + 3.46i)6-s + (10.4 + 6.03i)7-s − 15·8-s + (5.50 + 9.52i)9-s + (−0.982 + 0.567i)10-s + (−12.4 − 21.6i)11-s + 28·12-s + (49.9 + 28.8i)13-s + (−10.4 + 6.03i)14-s + (3.92 − 2.26i)15-s + (−20.5 + 35.5i)16-s + (96.3 − 55.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.176 + 0.306i)2-s + (0.384 − 0.666i)3-s + (0.437 + 0.757i)4-s + (0.0878 + 0.0507i)5-s + (0.136 + 0.235i)6-s + (0.564 + 0.325i)7-s − 0.662·8-s + (0.203 + 0.352i)9-s + (−0.0310 + 0.0179i)10-s + (−0.342 − 0.592i)11-s + 0.673·12-s + (1.06 + 0.614i)13-s + (−0.199 + 0.115i)14-s + (0.0676 − 0.0390i)15-s + (−0.320 + 0.554i)16-s + (1.37 − 0.793i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.76980 + 0.630839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76980 + 0.630839i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 + (-621. + 725. i)T \) |
| good | 2 | \( 1 + (0.5 - 0.866i)T + (-4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-2 + 3.46i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (-0.982 - 0.567i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-10.4 - 6.03i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (12.4 + 21.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-49.9 - 28.8i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-96.3 + 55.6i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 - 98.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (38.0 - 21.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-55.0 - 31.7i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (42.8 - 74.2i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (363. + 210. i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (209. + 120. i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-231. + 401. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 328.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-58.9 + 102. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (249. + 143. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-76.3 + 132. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 - 899. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + (-14.1 + 8.17i)T + (1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (408. - 707. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 39.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-715. + 413. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
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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
−13.70612091930937170125019993025, −12.42674789524452169534402050135, −11.71296093376819956289248052995, −10.40498820207924090098762402871, −8.644722133592956943309885372429, −8.030215997347656041600303988196, −7.01709261773556322872601797340, −5.63509862821192293263431686574, −3.50417371124120395350634452152, −1.88377419722512242022034212304,
1.34577867981263116491605203601, 3.32052905352415152248387237935, 4.94761146252885544825803118870, 6.32742313330055159721289787359, 7.895483047854451039788831638280, 9.251521270607732902972552381415, 10.18323281947056171138671539257, 10.88452977518360312129093292698, 12.10396720831428148817651785045, 13.44287927788870927014629321746
