| L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (−0.959 + 0.281i)6-s + (−2.22 − 1.43i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.856 + 5.95i)11-s + (0.142 + 0.989i)12-s + (−5.09 + 3.27i)13-s + (−1.73 + 2.00i)14-s + (0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (−1.34 + 0.395i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.292 + 0.337i)5-s + (−0.391 + 0.115i)6-s + (−0.841 − 0.540i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.266 − 0.170i)10-s + (0.258 + 1.79i)11-s + (0.0410 + 0.285i)12-s + (−1.41 + 0.908i)13-s + (−0.463 + 0.534i)14-s + (0.107 − 0.234i)15-s + (0.210 + 0.135i)16-s + (−0.326 + 0.0959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.473600 + 0.313311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.473600 + 0.313311i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.00 + 2.63i)T \) |
| good | 7 | \( 1 + (2.22 + 1.43i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.856 - 5.95i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (5.09 - 3.27i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.34 - 0.395i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.44 - 0.424i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.48 + 0.436i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.799 + 1.75i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.68 - 6.56i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.73 - 2.00i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.80 - 8.33i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 0.964T + 47T^{2} \) |
| 53 | \( 1 + (6.68 + 4.29i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-5.92 + 3.80i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (6.02 - 13.1i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.0565 - 0.393i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.49 + 10.4i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-5.18 - 1.52i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-10.1 + 6.52i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (7.80 - 9.00i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.651 + 1.42i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (6.18 + 7.14i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.47833208977272498889245207695, −9.805727810874638438221952781332, −9.410678707013326726058283804041, −7.86856733541841039611173381670, −6.94669852411288200657392076610, −6.45059092790516067343222587425, −4.93710105504606850645241045476, −4.16379350701685473557350582781, −2.68494336897341343293954173067, −1.76535170495995472841063842934,
0.28439735182725302788735873648, 2.81044625451680112369598553587, 3.79306198767572738414153403986, 5.22408329930894639014819722493, 5.67829992697758662392842512192, 6.53835917609314350230615292197, 7.71781292517322653268797560149, 8.704558551467356509513109477785, 9.335525431592162084105092613753, 10.10298356222080448010137598605
