L(s) = 1 | + (−2.66 + 0.933i)2-s + (−3.69 + 3.69i)3-s + (6.25 − 4.98i)4-s + (6.41 − 13.3i)6-s + (19.3 + 19.3i)7-s + (−12.0 + 19.1i)8-s − 0.287i·9-s + 23.4i·11-s + (−4.69 + 41.5i)12-s + (−62.9 − 62.9i)13-s + (−69.7 − 33.5i)14-s + (14.2 − 62.3i)16-s + (0.872 − 0.872i)17-s + (0.268 + 0.768i)18-s − 57.0·19-s + ⋯ |
L(s) = 1 | + (−0.943 + 0.330i)2-s + (−0.710 + 0.710i)3-s + (0.782 − 0.623i)4-s + (0.436 − 0.905i)6-s + (1.04 + 1.04i)7-s + (−0.532 + 0.846i)8-s − 0.0106i·9-s + 0.641i·11-s + (−0.112 + 0.998i)12-s + (−1.34 − 1.34i)13-s + (−1.33 − 0.641i)14-s + (0.223 − 0.974i)16-s + (0.0124 − 0.0124i)17-s + (0.00351 + 0.0100i)18-s − 0.688·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0185i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.00437468 - 0.472028i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00437468 - 0.472028i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.66 - 0.933i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (3.69 - 3.69i)T - 27iT^{2} \) |
| 7 | \( 1 + (-19.3 - 19.3i)T + 343iT^{2} \) |
| 11 | \( 1 - 23.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (62.9 + 62.9i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-0.872 + 0.872i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 57.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + (116. - 116. i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 121. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 66.8iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-36.6 + 36.6i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 302.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-190. + 190. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-54.1 - 54.1i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-36.5 - 36.5i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + 401.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 509.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-187. - 187. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 584. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-436. - 436. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 608.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-124. + 124. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 684. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (384. - 384. i)T - 9.12e5iT^{2} \) |
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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
−14.45745103819044231137434932491, −12.39828638698897946266493203374, −11.55880290032207310933026463699, −10.51380993890187460371039589476, −9.764128217121988856117337206339, −8.424113398151205012364050754739, −7.45331631279708008575024765328, −5.65439997333687831548906347586, −4.99703770796657868972886924781, −2.17758082336768653725211242440,
0.37866571276377995695018880138, 1.91456050823536014012144313007, 4.30764699513120455125892631575, 6.36021795638152970970252436772, 7.28121247079804753799039581943, 8.286110123872748800252744954997, 9.717327674652725007814823438412, 10.88133427725649815557158082052, 11.65593287169540832353507186289, 12.40714409616324157188509335756