| L(s) = 1 | + (−0.284 − 1.39i)2-s + (0.987 + 0.160i)3-s + (−0.0274 + 0.0117i)4-s + (4.22 − 1.04i)5-s + (−0.0573 − 1.42i)6-s + (−1.74 + 3.67i)7-s + (−1.59 − 2.31i)8-s + (0.948 + 0.316i)9-s + (−2.65 − 5.59i)10-s + (1.76 − 0.587i)11-s + (−0.0290 + 0.00714i)12-s + (−2.68 − 2.40i)13-s + (5.62 + 1.38i)14-s + (4.33 − 0.349i)15-s + (−2.81 + 2.92i)16-s + (1.16 − 2.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.201 − 0.987i)2-s + (0.569 + 0.0926i)3-s + (−0.0137 + 0.00585i)4-s + (1.88 − 0.465i)5-s + (−0.0234 − 0.581i)6-s + (−0.658 + 1.38i)7-s + (−0.563 − 0.816i)8-s + (0.316 + 0.105i)9-s + (−0.839 − 1.76i)10-s + (0.530 − 0.177i)11-s + (−0.00837 + 0.00206i)12-s + (−0.744 − 0.667i)13-s + (1.50 + 0.370i)14-s + (1.11 − 0.0903i)15-s + (−0.702 + 0.731i)16-s + (0.283 − 0.597i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64703 - 1.30137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64703 - 1.30137i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.987 - 0.160i)T \) |
| 13 | \( 1 + (2.68 + 2.40i)T \) |
| good | 2 | \( 1 + (0.284 + 1.39i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (-4.22 + 1.04i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (1.74 - 3.67i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-1.76 + 0.587i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (-1.16 + 2.46i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-0.447 - 0.775i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.66 - 6.34i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.453 - 2.22i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-3.88 - 2.03i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (2.16 - 1.37i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (3.95 + 0.642i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (8.25 + 5.21i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (1.18 + 9.77i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-4.46 - 6.47i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-3.42 - 3.56i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (0.751 + 0.0606i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (9.51 + 4.05i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (8.82 - 10.8i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (3.47 + 3.07i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (-1.50 - 12.4i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-2.59 + 6.82i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (0.211 - 0.366i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.05 + 10.5i)T + (-81.9 + 51.8i)T^{2} \) |
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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
−10.15275607921084691485121712557, −10.01205882129728903837559862184, −9.193253587940284668361928839210, −8.672089203112376608896128229038, −6.89087859333442809419161209451, −5.89438370669054380565762036073, −5.25227344972162074365027627358, −3.23759494320974874854242322046, −2.46540963719254923326086880314, −1.56993600889962728589360222812,
1.86430338275902506353779736289, 2.99872279689961427028792366497, 4.56744775802173383358424395426, 6.05192701263327961790327298912, 6.61668740240329719487405591631, 7.15513056725837122438072305044, 8.310963408426529826845404107402, 9.426650416382400247007552453886, 9.943770591282801680600300169607, 10.67754191187304788548466089515
