| L(s) = 1 | − i·2-s + 4-s + (−2 + i)5-s − 2i·7-s − 3i·8-s + (1 + 2i)10-s − 2·11-s − 2i·13-s − 2·14-s − 16-s − 2i·17-s − 19-s + (−2 + i)20-s + 2i·22-s + (3 − 4i)25-s − 2·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s + (−0.894 + 0.447i)5-s − 0.755i·7-s − 1.06i·8-s + (0.316 + 0.632i)10-s − 0.603·11-s − 0.554i·13-s − 0.534·14-s − 0.250·16-s − 0.485i·17-s − 0.229·19-s + (−0.447 + 0.223i)20-s + 0.426i·22-s + (0.600 − 0.800i)25-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.248448 - 1.05244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.248448 - 1.05244i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−10.28861374002305199843058850931, −9.128616988427608653322791127689, −7.85720717402839161488270301940, −7.36596128272417927145266405831, −6.58309485197631130609406322773, −5.28509586863728655778752439984, −3.96246734792274788932724653669, −3.32898224247665400544275299909, −2.19232002144848551942770727194, −0.49308327442812810118920035591,
1.87447109145379178149974946641, 3.13914195681282117764075202651, 4.45431662823273891089033482771, 5.41736537745405259602797701921, 6.20532620623201522542730246763, 7.24480507230017162053532898912, 7.905911542458527280977634980468, 8.619296314052961747359287112903, 9.422755247394498177531821990707, 10.79825486131797391212962993649
