L(s) = 1 | + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (0.0700 + 0.153i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (5.28 − 1.55i)11-s + (−0.959 + 0.281i)12-s + (1.37 − 3.01i)13-s + (0.0239 + 0.166i)14-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (−3.71 + 2.38i)17-s + ⋯ |
L(s) = 1 | + (0.678 + 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (0.0636 − 0.442i)5-s + (−0.343 + 0.220i)6-s + (0.0264 + 0.0579i)7-s + (0.231 + 0.267i)8-s + (−0.0474 − 0.329i)9-s + (0.131 − 0.287i)10-s + (1.59 − 0.467i)11-s + (−0.276 + 0.0813i)12-s + (0.381 − 0.835i)13-s + (0.00641 + 0.0445i)14-s + (0.169 + 0.195i)15-s + (0.103 + 0.227i)16-s + (−0.901 + 0.579i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.16067 + 0.477264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16067 + 0.477264i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 5 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (-3.07 - 3.67i)T \) |
good | 7 | \( 1 + (-0.0700 - 0.153i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-5.28 + 1.55i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.37 + 3.01i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (3.71 - 2.38i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.75 - 2.41i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.33 + 2.14i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (2.47 + 2.85i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.34 - 9.36i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.37 - 9.53i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.12 + 8.21i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 + (4.59 + 10.0i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (1.51 - 3.32i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.28 - 3.79i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (13.1 + 3.84i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (9.05 + 2.65i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.80 - 5.65i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.83 + 12.7i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.07 + 7.46i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (0.489 - 0.564i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.17 - 8.17i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.67567889254225147657279133424, −9.658277369147363487158676729599, −8.823822024640470302652883086036, −7.937231377757989293527475882457, −6.64100525702543611741950121346, −5.99645431216064349677896081807, −5.08173268607138634414328275931, −4.07302740662840370958420638902, −3.25426513563246879150009286166, −1.34077086324487606727505555202,
1.35065658307288757789595438859, 2.63964894189498144237656238046, 3.98339880033579781950538280179, 4.80814292563364170671378745634, 6.07301465693912489875206918993, 6.82559456256049313018769726153, 7.27511542757619882299741926435, 8.920958209002146798990049285318, 9.477395319107332818764615196522, 10.89950639545826862277875638156