| L(s) = 1 | + 81i·3-s − 1.82e3i·5-s − 1.07e4·7-s − 6.56e3·9-s − 1.90e4i·11-s + 4.34e4i·13-s + 1.47e5·15-s + 2.47e5·17-s + 2.10e5i·19-s − 8.67e5i·21-s − 7.57e5·23-s − 1.38e6·25-s − 5.31e5i·27-s + 3.97e6i·29-s + 5.47e6·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.30i·5-s − 1.68·7-s − 0.333·9-s − 0.391i·11-s + 0.421i·13-s + 0.754·15-s + 0.717·17-s + 0.371i·19-s − 0.972i·21-s − 0.564·23-s − 0.706·25-s − 0.192i·27-s + 1.04i·29-s + 1.06·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.007475103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007475103\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 81iT \) |
| good | 5 | \( 1 + 1.82e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 + 1.07e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.90e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 4.34e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 2.47e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.10e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + 7.57e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.97e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 5.47e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.85e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 2.61e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.12e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 2.52e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.17e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 5.26e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 7.03e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 2.56e7iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 1.94e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.12e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 8.17e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.77e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 4.48e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.52e8T + 7.60e17T^{2} \) |
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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
−9.604231922423849432773365008506, −8.875676311229044459379501208899, −7.976061113022073982393899062897, −6.55410826480199600737647944674, −5.76985301871770721231522664883, −4.74305004720190292949335513996, −3.75596421156254891163855404964, −2.87533819366375378784657426627, −1.25403093302822948077901721523, −0.26632936684015839501259005994,
0.72201261381317915100526183891, 2.39108670944117837357513775303, 2.97911840817578118823881358202, 3.93338200883617380107246470758, 5.72320411517496616000030242506, 6.44029874864104080644998383478, 7.09595641843508527310909711881, 7.943534634897126738193511166128, 9.354858514399001521630396838591, 10.07065036493438052474504427269
