| L(s) = 1 | + (2.79 − 0.442i)2-s + (−4.80 − 1.98i)3-s + (7.60 − 2.47i)4-s + (−13.0 + 25.6i)5-s + (−14.2 − 3.43i)6-s + (83.6 − 20.0i)7-s + (20.1 − 10.2i)8-s + (−38.1 − 38.1i)9-s + (−25.1 + 77.4i)10-s + (179. + 153. i)11-s + (−41.4 − 3.26i)12-s + (91.6 + 149. i)13-s + (224. − 93.1i)14-s + (113. − 97.1i)15-s + (51.7 − 37.6i)16-s + (−12.7 − 162. i)17-s + ⋯ |
| L(s) = 1 | + (0.698 − 0.110i)2-s + (−0.533 − 0.220i)3-s + (0.475 − 0.154i)4-s + (−0.522 + 1.02i)5-s + (−0.396 − 0.0952i)6-s + (1.70 − 0.409i)7-s + (0.315 − 0.160i)8-s + (−0.471 − 0.471i)9-s + (−0.251 + 0.774i)10-s + (1.48 + 1.27i)11-s + (−0.287 − 0.0226i)12-s + (0.542 + 0.884i)13-s + (1.14 − 0.475i)14-s + (0.505 − 0.431i)15-s + (0.202 − 0.146i)16-s + (−0.0441 − 0.561i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.35004 + 0.235743i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.35004 + 0.235743i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.79 + 0.442i)T \) |
| 41 | \( 1 + (1.35e3 - 991. i)T \) |
| good | 3 | \( 1 + (4.80 + 1.98i)T + (57.2 + 57.2i)T^{2} \) |
| 5 | \( 1 + (13.0 - 25.6i)T + (-367. - 505. i)T^{2} \) |
| 7 | \( 1 + (-83.6 + 20.0i)T + (2.13e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (-179. - 153. i)T + (2.29e3 + 1.44e4i)T^{2} \) |
| 13 | \( 1 + (-91.6 - 149. i)T + (-1.29e4 + 2.54e4i)T^{2} \) |
| 17 | \( 1 + (12.7 + 162. i)T + (-8.24e4 + 1.30e4i)T^{2} \) |
| 19 | \( 1 + (-232. + 379. i)T + (-5.91e4 - 1.16e5i)T^{2} \) |
| 23 | \( 1 + (121. - 166. i)T + (-8.64e4 - 2.66e5i)T^{2} \) |
| 29 | \( 1 + (58.3 - 740. i)T + (-6.98e5 - 1.10e5i)T^{2} \) |
| 31 | \( 1 + (1.07e3 + 347. i)T + (7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (540. + 1.66e3i)T + (-1.51e6 + 1.10e6i)T^{2} \) |
| 43 | \( 1 + (925. - 146. i)T + (3.25e6 - 1.05e6i)T^{2} \) |
| 47 | \( 1 + (-256. - 61.5i)T + (4.34e6 + 2.21e6i)T^{2} \) |
| 53 | \( 1 + (-1.07e3 - 84.6i)T + (7.79e6 + 1.23e6i)T^{2} \) |
| 59 | \( 1 + (4.95e3 + 3.60e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (-114. + 721. i)T + (-1.31e7 - 4.27e6i)T^{2} \) |
| 67 | \( 1 + (-2.75e3 - 3.22e3i)T + (-3.15e6 + 1.99e7i)T^{2} \) |
| 71 | \( 1 + (-6.23e3 + 7.30e3i)T + (-3.97e6 - 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-641. + 641. i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + (-491. + 1.18e3i)T + (-2.75e7 - 2.75e7i)T^{2} \) |
| 83 | \( 1 + 3.14e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (3.77e3 - 906. i)T + (5.59e7 - 2.84e7i)T^{2} \) |
| 97 | \( 1 + (-1.53e3 + 1.31e3i)T + (1.38e7 - 8.74e7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.00953364469077756125699762593, −12.17968499320938409444629192678, −11.36176757699782010627743098676, −11.11394665561117746036765050089, −9.184970789279574424984247697716, −7.32645166263358715205543220006, −6.72368875042529668596653444602, −4.97748302933287448904830259741, −3.75763239479792575364130704931, −1.61539191263087499676357243165,
1.29987062036519568184654010969, 3.86961753831889068636831274540, 5.12788815362300338933459349073, 5.91057958551774657503873366326, 8.142218616489513045301405701950, 8.536418749142949484402741230759, 10.78375249744869372582906012315, 11.63200967949332975619296202521, 12.15836303423047543090030416977, 13.72472218111398442627986947715
