| L(s) = 1 | + i·5-s − 3.60i·7-s − i·11-s + 4·13-s + (−3.60 + 2i)17-s + 7·19-s − 25-s − 9i·29-s + 7.21i·31-s + 3.60·35-s − 3.60i·37-s + 5i·41-s − 2·43-s − 10.8·47-s − 5.99·49-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 1.36i·7-s − 0.301i·11-s + 1.10·13-s + (−0.874 + 0.485i)17-s + 1.60·19-s − 0.200·25-s − 1.67i·29-s + 1.29i·31-s + 0.609·35-s − 0.592i·37-s + 0.780i·41-s − 0.304·43-s − 1.57·47-s − 0.857·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.812196734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812196734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 17 | \( 1 + (3.60 - 2i)T \) |
| good | 7 | \( 1 + 3.60iT - 7T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 - 7.21iT - 31T^{2} \) |
| 37 | \( 1 + 3.60iT - 37T^{2} \) |
| 41 | \( 1 - 5iT - 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 14.4iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8iT - 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 7.21iT - 97T^{2} \) |
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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
−8.380172007962167692336296501796, −7.898844883272046176125769840436, −6.96087555950024539274824298373, −6.51608198337392298073308334309, −5.59758593285001821120463490856, −4.56790246824092573342639219723, −3.74003356393194628110889943261, −3.16892381833001943552503352834, −1.74607421110152453045737987739, −0.64915745649792418934194978111,
1.14859618952979730486750268422, 2.25047014565339984894246698994, 3.16638824158979440098255821801, 4.15511853916350327083170011893, 5.29407275213969366936365073209, 5.51877952544068842863396972827, 6.57721578110188042382650383215, 7.29480981629599529086575597760, 8.411469766336645954664050647474, 8.713144610233142487362320840119
