| L(s) = 1 | + (0.0713 − 0.997i)2-s + (−0.977 + 0.212i)3-s + (−0.989 − 0.142i)4-s + (−0.342 − 2.20i)5-s + (0.142 + 0.989i)6-s + (−0.448 + 0.244i)7-s + (−0.212 + 0.977i)8-s + (0.909 − 0.415i)9-s + (−2.22 + 0.184i)10-s + (−1.95 − 1.69i)11-s + (0.997 − 0.0713i)12-s + (2.14 − 3.92i)13-s + (0.212 + 0.464i)14-s + (0.804 + 2.08i)15-s + (0.959 + 0.281i)16-s + (−0.765 − 0.573i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (−0.564 + 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.153 − 0.988i)5-s + (0.0580 + 0.404i)6-s + (−0.169 + 0.0925i)7-s + (−0.0751 + 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.704 + 0.0582i)10-s + (−0.589 − 0.510i)11-s + (0.287 − 0.0205i)12-s + (0.594 − 1.08i)13-s + (0.0566 + 0.124i)14-s + (0.207 + 0.538i)15-s + (0.239 + 0.0704i)16-s + (−0.185 − 0.138i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.123901 + 0.379393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123901 + 0.379393i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 3 | \( 1 + (0.977 - 0.212i)T \) |
| 5 | \( 1 + (0.342 + 2.20i)T \) |
| 23 | \( 1 + (4.69 - 0.985i)T \) |
| good | 7 | \( 1 + (0.448 - 0.244i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (1.95 + 1.69i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.14 + 3.92i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (0.765 + 0.573i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.787 - 5.47i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (9.80 - 1.40i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.87 - 1.20i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.879 - 0.327i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-3.97 + 8.70i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.200 - 0.920i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (5.32 + 5.32i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.49 - 2.72i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-3.62 - 12.3i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.08 + 3.24i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.14 - 0.439i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (5.17 + 5.97i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.78 + 2.38i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (13.7 - 4.02i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.24 + 14.0i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (1.23 + 0.791i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.35 + 6.31i)T + (-73.3 + 63.5i)T^{2} \) |
| 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
−10.14036166448617311240532621729, −9.207770048555970394483000463238, −8.338695681394879872062669130735, −7.58054927589446038188670804254, −5.72732084181541599923860549437, −5.62526807325842346325442592404, −4.25497473673664017610954662702, −3.38455850865976860446303658823, −1.69204986543294533259354361515, −0.21760599602115390523983342195,
2.17580985804656896272405911441, 3.73419367621685994002359123731, 4.67544275038590070327132973185, 5.88891257961378408077278605108, 6.61631459101248387162750023087, 7.26002273508416344728621501311, 8.125398970020975247736142659604, 9.349896588959820637225475297305, 10.05296744248830088364763743315, 11.19508309348692913377937906957
