| L(s) = 1 | + (0.707 − 0.707i)5-s + 2i·13-s + 1.41·17-s − 1.00i·25-s − 1.41i·29-s − 1.41i·41-s + 49-s − 1.41·53-s − 2·61-s + (1.41 + 1.41i)65-s + 2i·73-s + (1.00 − 1.00i)85-s + 1.41i·89-s − 2i·97-s + 1.41i·101-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)5-s + 2i·13-s + 1.41·17-s − 1.00i·25-s − 1.41i·29-s − 1.41i·41-s + 49-s − 1.41·53-s − 2·61-s + (1.41 + 1.41i)65-s + 2i·73-s + (1.00 − 1.00i)85-s + 1.41i·89-s − 2i·97-s + 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.270134969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.270134969\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 + 2iT - T^{2} \) |
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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
−9.553858824657240101368288896079, −9.090499428393930836314391837302, −8.194245400958252722655658960203, −7.27150602456082908267575014598, −6.32838191959848265892489280876, −5.62038769339938934261416149294, −4.64928075559940795806477178620, −3.85913127206617898592903337975, −2.38797304877923060345285475303, −1.39531844209752677672521383740,
1.38947236109725984958811829432, 2.93347467469201182710604855232, 3.32645719166491216701691684480, 4.95525508818493987900826338056, 5.67128581410216422987891073893, 6.34617814767493698739957582905, 7.48661474104666128968710882523, 7.936472115288443522594499985502, 9.057505269634614069947100778300, 9.909754869060367955775021839298
