| L(s) = 1 | + (−0.00779 + 0.00501i)2-s + (−0.0396 + 0.275i)3-s + (−0.830 + 1.81i)4-s + (−0.959 + 0.281i)5-s + (−0.00107 − 0.00234i)6-s + (2.74 + 3.16i)7-s + (−0.00527 − 0.0366i)8-s + (2.80 + 0.823i)9-s + (0.00606 − 0.00700i)10-s + (−4.08 − 2.62i)11-s + (−0.468 − 0.301i)12-s + (1.56 − 1.80i)13-s + (−0.0372 − 0.0109i)14-s + (−0.0396 − 0.275i)15-s + (−2.61 − 3.02i)16-s + (−0.0972 − 0.212i)17-s + ⋯ |
| L(s) = 1 | + (−0.00551 + 0.00354i)2-s + (−0.0228 + 0.159i)3-s + (−0.415 + 0.909i)4-s + (−0.429 + 0.125i)5-s + (−0.000437 − 0.000958i)6-s + (1.03 + 1.19i)7-s + (−0.00186 − 0.0129i)8-s + (0.934 + 0.274i)9-s + (0.00191 − 0.00221i)10-s + (−1.23 − 0.791i)11-s + (−0.135 − 0.0869i)12-s + (0.434 − 0.500i)13-s + (−0.00995 − 0.00292i)14-s + (−0.0102 − 0.0711i)15-s + (−0.654 − 0.755i)16-s + (−0.0235 − 0.0516i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.832233 + 0.512567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.832233 + 0.512567i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.60 - 1.34i)T \) |
| good | 2 | \( 1 + (0.00779 - 0.00501i)T + (0.830 - 1.81i)T^{2} \) |
| 3 | \( 1 + (0.0396 - 0.275i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (-2.74 - 3.16i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (4.08 + 2.62i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.56 + 1.80i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0972 + 0.212i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.130 + 0.285i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.30 + 5.04i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.00 + 6.95i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.41 - 1.58i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-4.88 + 1.43i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.784 - 5.45i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 4.61T + 47T^{2} \) |
| 53 | \( 1 + (5.66 + 6.53i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (6.45 - 7.45i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.150 - 1.04i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (8.98 - 5.77i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (4.95 - 3.18i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-7.01 + 15.3i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (9.52 - 10.9i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-7.03 - 2.06i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.79 + 12.5i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (6.60 - 1.93i)T + (81.6 - 52.4i)T^{2} \) |
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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
−13.44822041781247644787697911950, −12.83761734760498841565426169243, −11.64569031841616977590045250540, −10.87860130371535440777036307026, −9.298877131431627200712675613977, −8.154418531962990135587520260967, −7.66071392817465947973777709814, −5.58623446125804288603286850511, −4.41641163302231716509313617116, −2.77541582985470306071576318741,
1.37730755535431388421405856735, 4.25191392321128966256276251301, 5.04336599058115444928302243818, 6.90449846580346228404300058093, 7.78862415347931107833760749247, 9.212601224925147597891044660995, 10.50283537418180023473289852592, 10.91939140094349917255990661541, 12.54974828843921177316057196135, 13.44169967655326691580558215204
