| L(s) = 1 | + (−1.39 + 0.235i)2-s + (−2.43 − 1.70i)3-s + (1.88 − 0.658i)4-s + (−4.10 − 0.358i)5-s + (3.79 + 1.80i)6-s + (−3.20 − 1.85i)7-s + (−2.47 + 1.36i)8-s + (1.99 + 5.46i)9-s + (5.80 − 0.467i)10-s + (0.183 − 0.684i)11-s + (−5.71 − 1.61i)12-s + (−1.23 − 1.76i)13-s + (4.90 + 1.82i)14-s + (9.36 + 7.85i)15-s + (3.13 − 2.48i)16-s + (1.32 + 0.483i)17-s + ⋯ |
| L(s) = 1 | + (−0.985 + 0.166i)2-s + (−1.40 − 0.983i)3-s + (0.944 − 0.329i)4-s + (−1.83 − 0.160i)5-s + (1.54 + 0.735i)6-s + (−1.21 − 0.699i)7-s + (−0.876 + 0.482i)8-s + (0.663 + 1.82i)9-s + (1.83 − 0.147i)10-s + (0.0553 − 0.206i)11-s + (−1.64 − 0.466i)12-s + (−0.342 − 0.488i)13-s + (1.31 + 0.487i)14-s + (2.41 + 2.02i)15-s + (0.783 − 0.621i)16-s + (0.322 + 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0552576 + 0.0261292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0552576 + 0.0261292i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.235i)T \) |
| 19 | \( 1 + (0.525 + 4.32i)T \) |
| good | 3 | \( 1 + (2.43 + 1.70i)T + (1.02 + 2.81i)T^{2} \) |
| 5 | \( 1 + (4.10 + 0.358i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (3.20 + 1.85i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.183 + 0.684i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (1.23 + 1.76i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.32 - 0.483i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.66 + 1.98i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (3.64 - 7.80i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-0.0429 + 0.0743i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.726 + 0.726i)T + 37iT^{2} \) |
| 41 | \( 1 + (7.68 - 1.35i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.14 - 0.450i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (1.50 - 0.547i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (0.542 + 6.19i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (2.39 + 5.14i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-2.28 + 0.199i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (4.01 - 8.60i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (2.87 + 3.42i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (4.37 - 0.772i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.33 + 7.59i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.45 - 5.44i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-6.27 - 1.10i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-12.7 - 4.63i)T + (74.3 + 62.3i)T^{2} \) |
| 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
−11.68635676075447328572527801650, −11.01603365301963020295714233687, −10.31185671827135635037268553922, −8.822560145511096666292458607509, −7.65282270600356920576057879408, −7.12064382286091107936982215772, −6.46427602999113615018410586270, −5.05635662923288211165956591147, −3.31964051728354906132593212815, −0.77755748259711034385234606742,
0.11414360453533086020809122331, 3.29092425962547993485343208368, 4.22012163299313355477326363618, 5.79067566305648062351034612386, 6.75504042931722907086677948443, 7.74094801427692208303064245096, 9.032652485235410970663624991798, 9.857976603686558117453207794800, 10.65287838306630480611582637404, 11.65345736818991675455195736176
