| L(s) = 1 | + (0.306 + 1.70i)3-s − 2.23i·5-s + 2.64·7-s + (−2.81 + 1.04i)9-s − 5.55i·11-s + 7.13·13-s + (3.81 − 0.686i)15-s + 5.75i·17-s + (0.811 + 4.51i)21-s − 5.00·25-s + (−2.64 − 4.47i)27-s + 4.83i·29-s + (9.47 − 1.70i)33-s − 5.91i·35-s + (2.18 + 12.1i)39-s + ⋯ |
| L(s) = 1 | + (0.177 + 0.984i)3-s − 0.999i·5-s + 0.999·7-s + (−0.937 + 0.348i)9-s − 1.67i·11-s + 1.97·13-s + (0.984 − 0.177i)15-s + 1.39i·17-s + (0.177 + 0.984i)21-s − 1.00·25-s + (−0.509 − 0.860i)27-s + 0.898i·29-s + (1.64 − 0.296i)33-s − 0.999i·35-s + (0.350 + 1.94i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63635 + 0.146083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63635 + 0.146083i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.306 - 1.70i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
| good | 11 | \( 1 + 5.55iT - 11T^{2} \) |
| 13 | \( 1 - 7.13T + 13T^{2} \) |
| 17 | \( 1 - 5.75iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4.83iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 1.28iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 3.45T + 97T^{2} \) |
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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
−10.98278127206921195784034939421, −10.62765853478476798994862564096, −9.117640155632762363481926932737, −8.414404463608398678164195636619, −8.248558995293850894864109142022, −6.04544902668575513856251886633, −5.47464456821549268744404857370, −4.23822065682666598703026705078, −3.45901512393211054364822326427, −1.35558651723945636263736610204,
1.56909887509854704999650985035, 2.71147527760222378795449575142, 4.18148192257053158488706238799, 5.62942824195970461474153889946, 6.70981402939148616753787072234, 7.39525581579819809682380357749, 8.185011555928725432674966056272, 9.266926186028246335632643453304, 10.42693183888530306996969882416, 11.43148557092663401270070169917
