| L(s) = 1 | + (−1.61 + 1.17i)2-s + (0.927 − 2.85i)3-s + (1.23 − 3.80i)4-s + (−3.72 + 10.5i)5-s + (1.85 + 5.70i)6-s + 7.73·7-s + (2.47 + 7.60i)8-s + (−7.28 − 5.29i)9-s + (−6.37 − 21.4i)10-s + (−8.25 + 6.00i)11-s + (−9.70 − 7.05i)12-s + (−32.3 − 23.5i)13-s + (−12.5 + 9.09i)14-s + (26.6 + 20.3i)15-s + (−12.9 − 9.40i)16-s + (37.6 + 115. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.332 + 0.942i)5-s + (0.126 + 0.388i)6-s + 0.417·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.201 − 0.677i)10-s + (−0.226 + 0.164i)11-s + (−0.233 − 0.169i)12-s + (−0.690 − 0.501i)13-s + (−0.238 + 0.173i)14-s + (0.458 + 0.351i)15-s + (−0.202 − 0.146i)16-s + (0.537 + 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.506302 + 0.755087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.506302 + 0.755087i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 - 1.17i)T \) |
| 3 | \( 1 + (-0.927 + 2.85i)T \) |
| 5 | \( 1 + (3.72 - 10.5i)T \) |
| good | 7 | \( 1 - 7.73T + 343T^{2} \) |
| 11 | \( 1 + (8.25 - 6.00i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (32.3 + 23.5i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-37.6 - 115. i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-36.2 - 111. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (69.9 - 50.7i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (44.9 - 138. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-73.3 - 225. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (105. + 76.6i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-157. - 114. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 193.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (15.6 - 48.0i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-127. + 392. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-413. - 300. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-639. + 464. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (172. + 531. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (16.6 - 51.2i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-358. + 260. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-36.2 + 111. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (93.1 + 286. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (959. - 697. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-65.7 + 202. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.75721440412295115601755469436, −11.84016402134632223773387983529, −10.61443493223387030396043045673, −9.944806768486370132808232779375, −8.263443305154068163558279714326, −7.75521231854936540635807397854, −6.67100037436265141130207168008, −5.50771188256495835601126014379, −3.47516299108681845858089918949, −1.75031635394400088137280190157,
0.51775916447468740014259136599, 2.54251902550737608893983255800, 4.26914871409705391199190832763, 5.24407400102566501788303138951, 7.27669542000503866446589325764, 8.258254507189484937021939591644, 9.285864954462632778319580288467, 9.912782880301713695752282232240, 11.46291455414893163311670957721, 11.79670168303764992241998015091
