L(s) = 1 | + (1.85 + 3.54i)2-s + (−9.11 + 13.1i)4-s + (14.8 − 25.7i)5-s + (−51.8 + 29.9i)7-s + (−63.5 − 7.86i)8-s + (118. + 4.89i)10-s + (−195. + 112. i)11-s + (−85.8 + 148. i)13-s + (−202. − 128. i)14-s + (−90.0 − 239. i)16-s + 99.0·17-s − 169. i·19-s + (203. + 430. i)20-s + (−761. − 482. i)22-s + (−310. − 179. i)23-s + ⋯ |
L(s) = 1 | + (0.464 + 0.885i)2-s + (−0.569 + 0.822i)4-s + (0.595 − 1.03i)5-s + (−1.05 + 0.611i)7-s + (−0.992 − 0.122i)8-s + (1.18 + 0.0489i)10-s + (−1.61 + 0.931i)11-s + (−0.508 + 0.880i)13-s + (−1.03 − 0.654i)14-s + (−0.351 − 0.936i)16-s + 0.342·17-s − 0.468i·19-s + (0.508 + 1.07i)20-s + (−1.57 − 0.997i)22-s + (−0.587 − 0.339i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.134143 - 0.694258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134143 - 0.694258i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.85 - 3.54i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-14.8 + 25.7i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (51.8 - 29.9i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (195. - 112. i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (85.8 - 148. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 99.0T + 8.35e4T^{2} \) |
| 19 | \( 1 + 169. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (310. + 179. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (9.01 + 15.6i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-671. - 387. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 609.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (206. - 357. i)T + (-1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-265. + 153. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (2.27e3 - 1.31e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 2.03e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-2.25e3 - 1.29e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-708. - 1.22e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-5.19e3 - 2.99e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 1.23e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.06e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.63e3 + 944. i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-5.83e3 + 3.36e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + 9.43e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-7.29e3 - 1.26e4i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−13.34751133059505352378462307344, −12.82584613543992457803311512090, −12.07451401264143290739583330521, −9.926823800874568782594063262408, −9.222500841688552996805327367286, −8.030121021157372178299110242436, −6.71334066482228342161132848186, −5.49981697668757930772143574578, −4.62220748745947842467587554779, −2.60894074963938649029885388312,
0.25021714386292857569754764273, 2.61121561219941986056569473120, 3.42330470343126209978437382024, 5.40370043616163145173640950184, 6.38762050025639420550312761914, 7.977020281689785255248826157406, 9.917431601028097806674600907838, 10.20505233442878945284149631999, 11.13594412923587910280050230436, 12.62873812538706007991717451103