| L(s) = 1 | + (2.04 + 0.235i)2-s + (1.13 + 0.826i)3-s + (2.19 + 0.509i)4-s + (−0.564 + 0.825i)5-s + (2.13 + 1.95i)6-s + (3.19 + 3.29i)7-s + (0.486 + 0.173i)8-s + (−0.316 − 0.974i)9-s + (−1.35 + 1.55i)10-s + (−1.40 − 3.00i)11-s + (2.07 + 2.39i)12-s + (−3.49 + 5.79i)13-s + (5.78 + 7.49i)14-s + (−1.32 + 0.472i)15-s + (−3.08 − 1.51i)16-s + (2.89 − 2.36i)17-s + ⋯ |
| L(s) = 1 | + (1.44 + 0.166i)2-s + (0.656 + 0.476i)3-s + (1.09 + 0.254i)4-s + (−0.252 + 0.369i)5-s + (0.871 + 0.799i)6-s + (1.20 + 1.24i)7-s + (0.172 + 0.0613i)8-s + (−0.105 − 0.324i)9-s + (−0.426 + 0.492i)10-s + (−0.423 − 0.905i)11-s + (0.597 + 0.690i)12-s + (−0.969 + 1.60i)13-s + (1.54 + 2.00i)14-s + (−0.341 + 0.121i)15-s + (−0.770 − 0.379i)16-s + (0.702 − 0.574i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.27586 + 1.79971i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.27586 + 1.79971i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (1.40 + 3.00i)T \) |
| good | 2 | \( 1 + (-2.04 - 0.235i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (-1.13 - 0.826i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (-3.19 - 3.29i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (3.49 - 5.79i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-2.89 + 2.36i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-2.45 - 0.140i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 8.52i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (-0.459 + 1.18i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (-6.48 - 2.74i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (-0.343 + 3.99i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (10.3 - 5.85i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (0.574 - 0.168i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (0.685 - 3.38i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (-6.30 + 3.09i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (-2.89 - 1.63i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (3.74 - 0.430i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (2.60 - 5.71i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (2.30 + 8.77i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (-2.88 - 5.45i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (0.137 + 4.82i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (8.20 + 1.41i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (1.01 + 0.652i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (3.18 + 4.66i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−11.23559330056125889113912737109, −9.859737385462183626078452731974, −8.884880719482459781007768726493, −8.276953972103389154551351454110, −6.96794394154243664993383770421, −6.01134343770512582905597977774, −4.98187454292604506937099562129, −4.40370585294091106487287533333, −3.08344958507764506658988795901, −2.45463774803984865645780172331,
1.55116830812035188931452081818, 2.86951751160562396444349662317, 3.88699415436616462397174722924, 5.11287549658566525581115305188, 5.24162533208559819273908375542, 7.16877224530051930693355940818, 7.70498528256772445283540984012, 8.309009243619330584556256954456, 9.934483634672368306282108154299, 10.66895996608455834258492565340
