| L(s) = 1 | + (2.87 − 4.97i)2-s + (8.30 + 3.46i)3-s + (−8.51 − 14.7i)4-s + (24.3 + 5.74i)5-s + (41.1 − 31.3i)6-s + (−41.4 − 23.9i)7-s − 5.92·8-s + (56.9 + 57.6i)9-s + (98.5 − 104. i)10-s + (−73.2 − 42.2i)11-s + (−19.5 − 152. i)12-s + (−193. + 111. i)13-s + (−238. + 137. i)14-s + (182. + 132. i)15-s + (119. − 206. i)16-s − 434.·17-s + ⋯ |
| L(s) = 1 | + (0.718 − 1.24i)2-s + (0.922 + 0.385i)3-s + (−0.532 − 0.921i)4-s + (0.973 + 0.229i)5-s + (1.14 − 0.871i)6-s + (−0.845 − 0.487i)7-s − 0.0925·8-s + (0.702 + 0.711i)9-s + (0.985 − 1.04i)10-s + (−0.605 − 0.349i)11-s + (−0.135 − 1.05i)12-s + (−1.14 + 0.660i)13-s + (−1.21 + 0.701i)14-s + (0.809 + 0.587i)15-s + (0.465 − 0.806i)16-s − 1.50·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.34907 - 1.60313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34907 - 1.60313i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.30 - 3.46i)T \) |
| 5 | \( 1 + (-24.3 - 5.74i)T \) |
| good | 2 | \( 1 + (-2.87 + 4.97i)T + (-8 - 13.8i)T^{2} \) |
| 7 | \( 1 + (41.4 + 23.9i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (73.2 + 42.2i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (193. - 111. i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + 434.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 378.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-326. - 566. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (430. + 248. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-151. - 262. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 55.6iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-411. + 237. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (707. + 408. i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-435. + 753. i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 2.33e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (1.01e3 - 588. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-3.45e3 + 5.98e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.35e3 + 1.93e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 5.82e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 6.44e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (2.44e3 - 4.23e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (3.25e3 - 5.64e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + 1.38e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.16e4 - 6.74e3i)T + (4.42e7 + 7.66e7i)T^{2} \) |
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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
−14.28185994816800617519230489661, −13.51497160615806488731539733808, −12.92485834256347155394045280329, −11.19173718824359503410672823080, −10.03482847102890605233084044210, −9.365396157164196868897201657928, −7.16904269329065683417058064019, −4.94949224233529713549177344203, −3.33532481142731249782305426077, −2.16556499183514916530016279322,
2.60605378969782882003718638271, 4.91993861351249030071450374584, 6.33986832834333761090560744239, 7.41603338137272613775806609986, 8.898425952008286434229766863013, 10.04805771373600066370198253980, 12.76987248561940118424403658015, 13.10441336238746113111724343147, 14.26538155306484801406837477148, 15.17097236221495509200472696224
