L(s) = 1 | + (0.258 + 0.965i)3-s + (1.15 − 2.38i)7-s + (−0.866 + 0.499i)9-s + (0.366 − 0.633i)11-s + (−0.707 + 0.707i)13-s + (3.15 − 0.845i)17-s + (−0.767 − 1.33i)19-s + (2.59 + 0.500i)21-s + (−0.138 + 0.517i)23-s + (−0.707 − 0.707i)27-s − 6.92i·29-s + (1.73 + i)31-s + (0.707 + 0.189i)33-s + (0.776 + 0.208i)37-s + (−0.866 − 0.500i)39-s + ⋯ |
L(s) = 1 | + (0.149 + 0.557i)3-s + (0.436 − 0.899i)7-s + (−0.288 + 0.166i)9-s + (0.110 − 0.191i)11-s + (−0.196 + 0.196i)13-s + (0.765 − 0.205i)17-s + (−0.176 − 0.305i)19-s + (0.566 + 0.109i)21-s + (−0.0289 + 0.107i)23-s + (−0.136 − 0.136i)27-s − 1.28i·29-s + (0.311 + 0.179i)31-s + (0.123 + 0.0329i)33-s + (0.127 + 0.0342i)37-s + (−0.138 − 0.0800i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.862376418\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.862376418\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.15 + 2.38i)T \) |
good | 11 | \( 1 + (-0.366 + 0.633i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.15 + 0.845i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.767 + 1.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.138 - 0.517i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + (-1.73 - i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.776 - 0.208i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 6.19iT - 41T^{2} \) |
| 43 | \( 1 + (-5.27 - 5.27i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.0507 + 0.189i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.08 + 1.36i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-6.36 + 11.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.42 + 3.13i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.81 + 6.76i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-1.39 - 5.20i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.96 + 2.86i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.896 - 0.896i)T - 83iT^{2} \) |
| 89 | \( 1 + (5.36 + 9.29i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.39 - 7.39i)T + 97iT^{2} \) |
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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
−9.115066759140712087443131858543, −8.207897478267134067476791795100, −7.61182289776789970767076041053, −6.76266022768060326166744205637, −5.78686474160795942281377545691, −4.87894278535561385670222631409, −4.15174860343598203597418168615, −3.35430790404795764557017811101, −2.16375054847577028412571699548, −0.72970171589273981011413560994,
1.20407685323599104535801054148, 2.26288861632155263591488806445, 3.14508449324439990405679209345, 4.31393882000371519829614551117, 5.38081730773377953627162376662, 5.90968712134103550417087335950, 6.92400895792692764548174058803, 7.64741002455217923961314375381, 8.426538198727810886941687475654, 8.959746432976275909544346769068