L(s) = 1 | − 10.3·2-s + (−12.6 − 9.17i)3-s + 75.7·4-s − 9.35i·5-s + (130. + 95.2i)6-s + 67.1·7-s − 454.·8-s + (74.6 + 231. i)9-s + 97.0i·10-s − 337.·11-s + (−955. − 695. i)12-s + 682. i·13-s − 697.·14-s + (−85.8 + 117. i)15-s + 2.29e3·16-s − 513. i·17-s + ⋯ |
L(s) = 1 | − 1.83·2-s + (−0.808 − 0.588i)3-s + 2.36·4-s − 0.167i·5-s + (1.48 + 1.08i)6-s + 0.518·7-s − 2.51·8-s + (0.306 + 0.951i)9-s + 0.307i·10-s − 0.841·11-s + (−1.91 − 1.39i)12-s + 1.11i·13-s − 0.950·14-s + (−0.0984 + 0.135i)15-s + 2.24·16-s − 0.431i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.02279274396\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02279274396\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (12.6 + 9.17i)T \) |
| 59 | \( 1 + (2.56e4 - 7.54e3i)T \) |
good | 2 | \( 1 + 10.3T + 32T^{2} \) |
| 5 | \( 1 + 9.35iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 67.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 337.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 682. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 513. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 189.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 784.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.83e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.99e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 9.26e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.65e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 5.28e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.08e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.72e3iT - 4.18e8T^{2} \) |
| 61 | \( 1 + 2.66e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.64e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 8.35e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 1.58e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5iT - 8.58e9T^{2} \) |
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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
−11.11785020299129855925395539960, −10.27649425677571960019033258531, −9.165197813626776974068160161616, −8.139205527482797328665480571879, −7.30932898981344696895584403798, −6.43349926839976692215331991562, −5.00457784685578759479125970491, −2.35048477536237204793226635131, −1.24133026726990805756373881732, −0.01776572594492171296601643852,
1.21712781070802172949687599879, 2.97919135533462185301286692197, 5.08902107977332946699712160522, 6.30501326415706515973057389883, 7.48910880146651507996641643946, 8.384908862735737768797915661217, 9.461340616229949435909506087475, 10.53933104824886174833668423510, 10.73651608300861565191800566380, 11.79614409806896592971221489867