L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.258 + 0.965i)3-s + (−0.499 + 0.866i)4-s + (1.50 + 1.64i)5-s + (−0.965 + 0.258i)6-s + (2.70 + 1.56i)7-s − 0.999·8-s + (−0.866 − 0.499i)9-s + (−0.674 + 2.13i)10-s + (0.628 + 0.168i)11-s + (−0.707 − 0.707i)12-s + (−1.78 − 3.13i)13-s + 3.12i·14-s + (−1.98 + 1.03i)15-s + (−0.5 − 0.866i)16-s + (−0.0844 + 0.0226i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.149 + 0.557i)3-s + (−0.249 + 0.433i)4-s + (0.674 + 0.737i)5-s + (−0.394 + 0.105i)6-s + (1.02 + 0.589i)7-s − 0.353·8-s + (−0.288 − 0.166i)9-s + (−0.213 + 0.674i)10-s + (0.189 + 0.0507i)11-s + (−0.204 − 0.204i)12-s + (−0.495 − 0.868i)13-s + 0.834i·14-s + (−0.512 + 0.266i)15-s + (−0.125 − 0.216i)16-s + (−0.0204 + 0.00548i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.911250 + 1.50670i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.911250 + 1.50670i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 + (-1.50 - 1.64i)T \) |
| 13 | \( 1 + (1.78 + 3.13i)T \) |
good | 7 | \( 1 + (-2.70 - 1.56i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.628 - 0.168i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (0.0844 - 0.0226i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.264 + 0.986i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.28 - 0.611i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (7.96 - 4.59i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.40 + 1.40i)T + 31iT^{2} \) |
| 37 | \( 1 + (-6.58 + 3.80i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.901 - 3.36i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.60 + 9.73i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + 4.74iT - 47T^{2} \) |
| 53 | \( 1 + (-6.10 - 6.10i)T + 53iT^{2} \) |
| 59 | \( 1 + (-12.7 + 3.42i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.40 + 5.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.42 + 9.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.0 + 3.48i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + 2.45T + 73T^{2} \) |
| 79 | \( 1 - 10.4iT - 79T^{2} \) |
| 83 | \( 1 - 1.51iT - 83T^{2} \) |
| 89 | \( 1 + (4.84 - 18.0i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.55 + 4.43i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.45046309149129984730374514329, −10.79798055121264065074247412477, −9.725347907922211248273427879553, −8.872111626214792444365956069114, −7.78966905044160206901524922309, −6.82441187806305476179649375773, −5.55929299993047037097199861950, −5.16626155258472896736905078836, −3.67105265435246911160078639283, −2.31348413534120937380356532112,
1.22034657004546872034158055556, 2.24175510572259094348932754427, 4.15673646950413166579085175416, 5.00484358810339014797609090814, 6.03198052846844560027686997912, 7.24255518133317387604698095867, 8.319231820985756571327621421436, 9.298896005982436314961246159366, 10.17758550548853571329451263482, 11.35088591041436467794699001276