| L(s) = 1 | + (−3.53 + 6.11i)3-s + (−1.05 + 0.611i)5-s + (16.5 − 8.40i)7-s + (−11.4 − 19.8i)9-s + (−3.96 − 2.28i)11-s + 41.0i·13-s − 8.64i·15-s + (−103. − 59.7i)17-s + (−30.0 − 52.1i)19-s + (−6.85 + 130. i)21-s + (−7.23 + 4.17i)23-s + (−61.7 + 106. i)25-s − 29.0·27-s − 164.·29-s + (143. − 249. i)31-s + ⋯ |
| L(s) = 1 | + (−0.679 + 1.17i)3-s + (−0.0948 + 0.0547i)5-s + (0.891 − 0.453i)7-s + (−0.423 − 0.733i)9-s + (−0.108 − 0.0627i)11-s + 0.876i·13-s − 0.148i·15-s + (−1.47 − 0.852i)17-s + (−0.363 − 0.629i)19-s + (−0.0712 + 1.35i)21-s + (−0.0655 + 0.0378i)23-s + (−0.494 + 0.855i)25-s − 0.207·27-s − 1.05·29-s + (0.833 − 1.44i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0742 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0742 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4570093172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4570093172\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-16.5 + 8.40i)T \) |
| good | 3 | \( 1 + (3.53 - 6.11i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (1.05 - 0.611i)T + (62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (3.96 + 2.28i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 41.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (103. + 59.7i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (30.0 + 52.1i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (7.23 - 4.17i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-143. + 249. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (74.3 + 128. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 358. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 360. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-112. - 195. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-342. + 592. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (42.5 - 73.6i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-181. + 104. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (361. + 208. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 982. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-295. - 170. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-66.0 + 38.1i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 523.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (841. - 486. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 676. iT - 9.12e5T^{2} \) |
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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
−10.64850353107866773121455248040, −9.606261868008207102370563068614, −8.942954893929143258863734938888, −7.67491945555567061569280900326, −6.67762991705548173750919880314, −5.41311768870653114707467489307, −4.56697241595566620507695463798, −3.95446597783833312652246082836, −2.11603055682549381615895626981, −0.16328116620057378058318144091,
1.33239282148337813697964459715, 2.35909788753779185826956389007, 4.19935171766082638189620472746, 5.44185739689888956610981296824, 6.19738869394541646705486842179, 7.16796384029149716389014633854, 8.123258514616604037593080791046, 8.727672012695010423500091074362, 10.30079344066083165029179469112, 11.03689305887166550157415172414
