| L(s) = 1 | + 3-s − 9i·5-s + 15i·7-s − 26·9-s − 48i·11-s + (26 − 39i)13-s − 9i·15-s − 45·17-s − 6i·19-s + 15i·21-s − 162·23-s + 44·25-s − 53·27-s − 144·29-s − 264i·31-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.804i·5-s + 0.809i·7-s − 0.962·9-s − 1.31i·11-s + (0.554 − 0.832i)13-s − 0.154i·15-s − 0.642·17-s − 0.0724i·19-s + 0.155i·21-s − 1.46·23-s + 0.351·25-s − 0.377·27-s − 0.922·29-s − 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.511235 - 0.955252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.511235 - 0.955252i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-26 + 39i)T \) |
| good | 3 | \( 1 - T + 27T^{2} \) |
| 5 | \( 1 + 9iT - 125T^{2} \) |
| 7 | \( 1 - 15iT - 343T^{2} \) |
| 11 | \( 1 + 48iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 45T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 162T + 1.21e4T^{2} \) |
| 29 | \( 1 + 144T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 303iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 192iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 97T + 7.95e4T^{2} \) |
| 47 | \( 1 - 111iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 414T + 1.48e5T^{2} \) |
| 59 | \( 1 - 522iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 376T + 2.26e5T^{2} \) |
| 67 | \( 1 - 36iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 357iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 830T + 4.93e5T^{2} \) |
| 83 | \( 1 - 438iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 438iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 852iT - 9.12e5T^{2} \) |
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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
−11.57075021157932089553411925028, −10.83767445112736452071249485639, −9.285853543463780898737790634313, −8.637012754573256213117197364097, −7.935356260737636540689566494178, −5.96672131472642785588565552657, −5.54962388904416021126833316409, −3.82127372866396789480956446674, −2.43361950037755137473745298590, −0.42847920676431061705164493210,
1.96302582713198543486559049877, 3.45012976609522425213677893327, 4.66310822948717736820270762512, 6.32406132983111856895754219442, 7.09432657121297879091367520974, 8.212538490795613286936716168173, 9.401945051072229739517571340721, 10.39508387755467811169448442890, 11.21354836158198665276144520867, 12.14853851309253719381556427669
