| L(s) = 1 | + (−0.342 + 0.939i)2-s + (2.77 + 1.29i)3-s + (−0.766 − 0.642i)4-s + (−0.312 − 2.21i)5-s + (−2.16 + 2.16i)6-s + (3.03 − 2.12i)7-s + (0.866 − 0.500i)8-s + (4.08 + 4.87i)9-s + (2.18 + 0.463i)10-s + (−4.99 + 2.88i)11-s + (−1.29 − 2.77i)12-s + (2.15 − 2.57i)13-s + (0.960 + 3.58i)14-s + (1.99 − 6.54i)15-s + (0.173 + 0.984i)16-s + (−1.13 + 0.951i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (1.60 + 0.746i)3-s + (−0.383 − 0.321i)4-s + (−0.139 − 0.990i)5-s + (−0.883 + 0.883i)6-s + (1.14 − 0.804i)7-s + (0.306 − 0.176i)8-s + (1.36 + 1.62i)9-s + (0.691 + 0.146i)10-s + (−1.50 + 0.869i)11-s + (−0.373 − 0.800i)12-s + (0.598 − 0.713i)13-s + (0.256 + 0.957i)14-s + (0.515 − 1.68i)15-s + (0.0434 + 0.246i)16-s + (−0.275 + 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81090 + 0.792861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81090 + 0.792861i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.342 - 0.939i)T \) |
| 5 | \( 1 + (0.312 + 2.21i)T \) |
| 37 | \( 1 + (4.28 - 4.31i)T \) |
| good | 3 | \( 1 + (-2.77 - 1.29i)T + (1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (-3.03 + 2.12i)T + (2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (4.99 - 2.88i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.15 + 2.57i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.13 - 0.951i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-2.94 - 1.37i)T + (12.2 + 14.5i)T^{2} \) |
| 23 | \( 1 + (-3.52 - 2.03i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.84 + 0.494i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (6.64 + 6.64i)T + 31iT^{2} \) |
| 41 | \( 1 + (7.07 - 8.43i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 - 1.19iT - 43T^{2} \) |
| 47 | \( 1 + (1.40 - 0.375i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (6.03 + 4.22i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (-6.30 + 9.00i)T + (-20.1 - 55.4i)T^{2} \) |
| 61 | \( 1 + (-0.0217 - 0.248i)T + (-60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (1.68 + 2.40i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-1.93 + 0.703i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.66 - 4.66i)T + 73iT^{2} \) |
| 79 | \( 1 + (5.89 - 4.12i)T + (27.0 - 74.2i)T^{2} \) |
| 83 | \( 1 + (-1.20 + 13.7i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-10.0 - 7.01i)T + (30.4 + 83.6i)T^{2} \) |
| 97 | \( 1 + (-6.51 + 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.19221145621887010288043204054, −10.19955964085788543960974697859, −9.547132962225108927317264567378, −8.472157526369573347109204026320, −7.936562932138059538379340474862, −7.48887840522774668385169469414, −5.22031224901991462137626150676, −4.68630818736697428699304794213, −3.54222383879629692451578157272, −1.75036636644103428242999622049,
1.85533433151358845310728018831, 2.71030846420053505566179081860, 3.55565652987748123900061993934, 5.30240383617286964990592952419, 7.00208505071686987070054758830, 7.75680443025291351694640721367, 8.622954192696408677849315358407, 9.044468410054537349395769729065, 10.52325560820509621886613499854, 11.18551679985314111163519643913
