| L(s) = 1 | + (−3.38 + 2.17i)2-s + (2.21 − 15.4i)3-s + (−19.8 + 43.4i)4-s + (2.95 + 10.0i)5-s + (26.0 + 57.1i)6-s + (209. − 181. i)7-s + (−64.0 − 445. i)8-s + (−233. − 68.4i)9-s + (−31.9 − 27.6i)10-s + (−64.3 + 100. i)11-s + (626. + 402. i)12-s + (−1.62e3 + 1.87e3i)13-s + (−314. + 1.07e3i)14-s + (161. − 23.2i)15-s + (−814. − 940. i)16-s + (3.65e3 − 1.67e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.423 + 0.272i)2-s + (0.0821 − 0.571i)3-s + (−0.310 + 0.679i)4-s + (0.0236 + 0.0805i)5-s + (0.120 + 0.264i)6-s + (0.610 − 0.528i)7-s + (−0.125 − 0.870i)8-s + (−0.319 − 0.0939i)9-s + (−0.0319 − 0.0276i)10-s + (−0.0483 + 0.0752i)11-s + (0.362 + 0.233i)12-s + (−0.740 + 0.854i)13-s + (−0.114 + 0.390i)14-s + (0.0479 − 0.00689i)15-s + (−0.198 − 0.229i)16-s + (0.744 − 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.28820 + 0.516898i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28820 + 0.516898i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.21 + 15.4i)T \) |
| 23 | \( 1 + (-6.40e3 - 1.03e4i)T \) |
| good | 2 | \( 1 + (3.38 - 2.17i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-2.95 - 10.0i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (-209. + 181. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (64.3 - 100. i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (1.62e3 - 1.87e3i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (-3.65e3 + 1.67e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (-1.03e4 - 4.71e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (-1.02e4 - 2.24e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (-5.39e3 - 3.74e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (8.54e3 - 2.90e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (-9.90e4 + 2.90e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (1.11e5 + 1.60e4i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 2.09e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.81e5 + 1.57e5i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (-1.07e5 + 1.24e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (3.17e5 - 4.57e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (1.34e4 + 2.09e4i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (3.27e5 - 2.10e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (1.85e4 - 4.06e4i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-4.99e5 - 4.32e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (-2.41e5 + 8.23e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (1.64e5 + 2.37e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (4.49e5 + 1.53e6i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−13.78333603707801649975437639678, −12.45385910302989502022664715223, −11.67713106855217278533287417824, −10.03519412448686445819813848936, −8.838491979828182939028539590178, −7.62890104667394048413658666314, −6.99410758626884150124075853819, −4.95032154390330556854137774799, −3.24380407454272726969916473465, −1.15296238393255777091363543595,
0.796961333461875696948219178562, 2.67335233385189519337360108116, 4.80887361248085325874926133796, 5.67718368726571963846546203005, 7.81966676633728565541618570263, 9.015248204914753971286186932676, 9.921228984242439829073198256765, 10.92113199760253083447106588751, 11.99716668311493030210390076744, 13.53027116008195265100183760006
