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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.194522136\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.194522136\) |
\(L(\frac{3}{2})\) |
|
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
−9.141285959920291445945374241649, −8.599089354947549090662402606827, −7.82187888359815979150142258144, −6.88076461897142697020163786290, −6.20825986995418307070239855908, −5.53146798779132581721841977921, −4.34698642159129579580448463864, −3.45390103377331033853857539157, −1.96723011217190135689600229666, −1.16449962113791533296203346808,
0.53205929186200631853239362562, 2.80259829706461570882448177630, 3.37890952185405450857573276259, 3.98268057301808885933504740169, 5.45424739768460286939897005091, 6.15378963792756776833691251937, 6.55293837315420006105816546224, 7.86045261963267113168232225515, 8.772973999834265909094023833312, 9.328940571871372331660770677247