L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.471 + 1.66i)3-s + 1.00i·4-s + (−1.55 + 1.60i)5-s + (0.844 − 1.51i)6-s + (−3.44 + 3.44i)7-s + (0.707 − 0.707i)8-s + (−2.55 + 1.57i)9-s + (2.23 − 0.0307i)10-s − 3.84i·11-s + (−1.66 + 0.471i)12-s + (0.875 + 0.875i)13-s + 4.87·14-s + (−3.40 − 1.84i)15-s − 1.00·16-s + (1.35 + 1.35i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.272 + 0.962i)3-s + 0.500i·4-s + (−0.697 + 0.716i)5-s + (0.344 − 0.617i)6-s + (−1.30 + 1.30i)7-s + (0.250 − 0.250i)8-s + (−0.851 + 0.524i)9-s + (0.707 − 0.00972i)10-s − 1.15i·11-s + (−0.481 + 0.136i)12-s + (0.242 + 0.242i)13-s + 1.30·14-s + (−0.879 − 0.475i)15-s − 0.250·16-s + (0.329 + 0.329i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0633502 - 0.237401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0633502 - 0.237401i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.471 - 1.66i)T \) |
| 5 | \( 1 + (1.55 - 1.60i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (3.44 - 3.44i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.84iT - 11T^{2} \) |
| 13 | \( 1 + (-0.875 - 0.875i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.35 - 1.35i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.81iT - 19T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 + (-1.86 + 1.86i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.29iT - 41T^{2} \) |
| 43 | \( 1 + (-0.142 - 0.142i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.63 - 3.63i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.62 - 3.62i)T - 53iT^{2} \) |
| 59 | \( 1 + 3.87T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + (2.77 - 2.77i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.15iT - 71T^{2} \) |
| 73 | \( 1 + (9.60 + 9.60i)T + 73iT^{2} \) |
| 79 | \( 1 - 17.1iT - 79T^{2} \) |
| 83 | \( 1 + (2.00 - 2.00i)T - 83iT^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + (3.36 - 3.36i)T - 97iT^{2} \) |
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
−10.99061551830290576577245723414, −10.05620605353470435548306179202, −9.225640293979085984482808671501, −8.739551155255895930517203991903, −7.81128038710012464313518557726, −6.46256334621746514671326834714, −5.68798992905342943814610737442, −4.14740077985037771007718099356, −3.12747413739106165973913214010, −2.74121681002084566018095543482,
0.15098509677344160224492870782, 1.43102606267923859833776664566, 3.28986613768604860754438743903, 4.32390581204461359222217234373, 5.75207475526736831728858851570, 6.76347834508615874576605656533, 7.45734403269996317863137636922, 7.88769853950880859769612605000, 9.073440952987312086903367475540, 9.738441215136290835626432888323