| L(s) = 1 | + (1.99 − 0.104i)2-s + (−5.07 + 6.26i)3-s + (−3.98 + 0.419i)4-s + (−10.9 − 2.26i)5-s + (−9.46 + 13.0i)6-s + (15.9 − 9.35i)7-s + (−23.6 + 3.75i)8-s + (−7.90 − 37.1i)9-s + (−22.0 − 3.36i)10-s + (24.3 + 5.17i)11-s + (17.6 − 27.1i)12-s + (29.1 − 14.8i)13-s + (30.9 − 20.3i)14-s + (69.7 − 57.1i)15-s + (−15.4 + 3.29i)16-s + (−16.7 + 43.7i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0369i)2-s + (−0.976 + 1.20i)3-s + (−0.498 + 0.0523i)4-s + (−0.979 − 0.202i)5-s + (−0.644 + 0.886i)6-s + (0.862 − 0.505i)7-s + (−1.04 + 0.165i)8-s + (−0.292 − 1.37i)9-s + (−0.698 − 0.106i)10-s + (0.667 + 0.141i)11-s + (0.423 − 0.652i)12-s + (0.622 − 0.317i)13-s + (0.589 − 0.388i)14-s + (1.20 − 0.983i)15-s + (−0.242 + 0.0514i)16-s + (−0.239 + 0.623i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.667798 - 0.386679i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.667798 - 0.386679i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (10.9 + 2.26i)T \) |
| 7 | \( 1 + (-15.9 + 9.35i)T \) |
| good | 2 | \( 1 + (-1.99 + 0.104i)T + (7.95 - 0.836i)T^{2} \) |
| 3 | \( 1 + (5.07 - 6.26i)T + (-5.61 - 26.4i)T^{2} \) |
| 11 | \( 1 + (-24.3 - 5.17i)T + (1.21e3 + 541. i)T^{2} \) |
| 13 | \( 1 + (-29.1 + 14.8i)T + (1.29e3 - 1.77e3i)T^{2} \) |
| 17 | \( 1 + (16.7 - 43.7i)T + (-3.65e3 - 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-3.73 + 35.5i)T + (-6.70e3 - 1.42e3i)T^{2} \) |
| 23 | \( 1 + (11.4 + 218. i)T + (-1.21e4 + 1.27e3i)T^{2} \) |
| 29 | \( 1 + (130. + 179. i)T + (-7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-38.2 - 85.8i)T + (-1.99e4 + 2.21e4i)T^{2} \) |
| 37 | \( 1 + (218. + 141. i)T + (2.06e4 + 4.62e4i)T^{2} \) |
| 41 | \( 1 + (-36.1 - 11.7i)T + (5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-119. + 119. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (340. - 130. i)T + (7.71e4 - 6.94e4i)T^{2} \) |
| 53 | \( 1 + (224. + 181. i)T + (3.09e4 + 1.45e5i)T^{2} \) |
| 59 | \( 1 + (392. + 435. i)T + (-2.14e4 + 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-650. - 585. i)T + (2.37e4 + 2.25e5i)T^{2} \) |
| 67 | \( 1 + (586. + 225. i)T + (2.23e5 + 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-783. + 569. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (96.8 + 149. i)T + (-1.58e5 + 3.55e5i)T^{2} \) |
| 79 | \( 1 + (128. - 289. i)T + (-3.29e5 - 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-129. - 819. i)T + (-5.43e5 + 1.76e5i)T^{2} \) |
| 89 | \( 1 + (-741. + 823. i)T + (-7.36e4 - 7.01e5i)T^{2} \) |
| 97 | \( 1 + (-35.9 + 226. i)T + (-8.68e5 - 2.82e5i)T^{2} \) |
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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
−11.95583115048023684385273986299, −11.18608451691648590923732454829, −10.44367345375708813088476170585, −9.071005862482601775216264889747, −8.152196483407548596378831222292, −6.40144734632431423821870812706, −5.10928050790918127037660439613, −4.34722212751535893543575531973, −3.75856272016217531664709493375, −0.36508346878130150013121490396,
1.34257482556512106426821620744, 3.60889845611392371008733093458, 4.98032359795220412107836473226, 5.91384173699896346604626647341, 7.03280531367928166600405101006, 8.070590272879412816223406255998, 9.190729167976256198074417697590, 11.17630694241959436138186052628, 11.66157934007288600125349288777, 12.27614285286908604501844255589
