| L(s) = 1 | + (1.42 + 2.27i)2-s + (0.724 − 2.07i)3-s + (−1.39 + 2.88i)4-s + (−6.69 − 1.52i)5-s + (5.74 − 1.31i)6-s + (−4.78 + 2.30i)7-s + (2.11 − 0.238i)8-s + (3.27 + 2.61i)9-s + (−6.09 − 17.4i)10-s + (−3.10 − 0.349i)11-s + (4.97 + 4.97i)12-s + (14.9 − 11.9i)13-s + (−12.0 − 7.57i)14-s + (−8.02 + 12.7i)15-s + (11.5 + 14.5i)16-s + (−19.1 + 19.1i)17-s + ⋯ |
| L(s) = 1 | + (0.714 + 1.13i)2-s + (0.241 − 0.690i)3-s + (−0.347 + 0.722i)4-s + (−1.33 − 0.305i)5-s + (0.956 − 0.218i)6-s + (−0.682 + 0.328i)7-s + (0.264 − 0.0298i)8-s + (0.363 + 0.290i)9-s + (−0.609 − 1.74i)10-s + (−0.281 − 0.0317i)11-s + (0.414 + 0.414i)12-s + (1.14 − 0.915i)13-s + (−0.861 − 0.541i)14-s + (−0.534 + 0.851i)15-s + (0.722 + 0.906i)16-s + (−1.12 + 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13928 + 0.462483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13928 + 0.462483i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (22.5 - 18.2i)T \) |
| good | 2 | \( 1 + (-1.42 - 2.27i)T + (-1.73 + 3.60i)T^{2} \) |
| 3 | \( 1 + (-0.724 + 2.07i)T + (-7.03 - 5.61i)T^{2} \) |
| 5 | \( 1 + (6.69 + 1.52i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 + (4.78 - 2.30i)T + (30.5 - 38.3i)T^{2} \) |
| 11 | \( 1 + (3.10 + 0.349i)T + (117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (-14.9 + 11.9i)T + (37.6 - 164. i)T^{2} \) |
| 17 | \( 1 + (19.1 - 19.1i)T - 289iT^{2} \) |
| 19 | \( 1 + (-3.26 + 1.14i)T + (282. - 225. i)T^{2} \) |
| 23 | \( 1 + (3.11 + 13.6i)T + (-476. + 229. i)T^{2} \) |
| 31 | \( 1 + (30.9 + 49.2i)T + (-416. + 865. i)T^{2} \) |
| 37 | \( 1 + (-23.4 + 2.64i)T + (1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (-29.0 - 29.0i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-28.3 - 17.8i)T + (802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (-5.42 + 48.1i)T + (-2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (-2.93 + 12.8i)T + (-2.53e3 - 1.21e3i)T^{2} \) |
| 59 | \( 1 - 11.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + (18.2 - 52.2i)T + (-2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (6.83 + 5.44i)T + (998. + 4.37e3i)T^{2} \) |
| 71 | \( 1 + (49.1 - 39.1i)T + (1.12e3 - 4.91e3i)T^{2} \) |
| 73 | \( 1 + (-33.0 + 52.6i)T + (-2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (-5.05 - 44.8i)T + (-6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (123. + 59.5i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (-38.7 - 61.6i)T + (-3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (-16.9 - 48.4i)T + (-7.35e3 + 5.86e3i)T^{2} \) |
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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
−16.44928336807855201941437637174, −15.72897126988792903741666482545, −14.95156764510463607958060818374, −13.14676265693164104621548812828, −12.84628278555042872567093593408, −10.91536363681713318376553660960, −8.390620663453847311804182573687, −7.48557432808695614250748570858, −6.07221163745282262389688776946, −4.08670171568366734569995492993,
3.44172461303241243960910841954, 4.29959478948468261868990050981, 7.20386649415626310636730727760, 9.271165678538823703999266964144, 10.79575669638434183916865149891, 11.58604407007979843881385770772, 12.88632198872024851596473646810, 14.09051550036481991710050497931, 15.67413588185734315712784137574, 16.16309691269335600609777226004
