| L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s − 6·11-s − i·13-s + 14-s + 16-s + 3i·17-s − 2·19-s − 6i·22-s + 26-s + i·28-s + 6·29-s − 4·31-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s − 1.80·11-s − 0.277i·13-s + 0.267·14-s + 0.250·16-s + 0.727i·17-s − 0.458·19-s − 1.27i·22-s + 0.196·26-s + 0.188i·28-s + 1.11·29-s − 0.718·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198091870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198091870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.309891891730272187527124136661, −7.41621943509078309743333663383, −7.01707475704313218658934537374, −5.94559411543696727951580661390, −5.52457448018684542945298567970, −4.70865249946854821548370099585, −3.97326531411485703370947086475, −2.99972213986979563729315769828, −2.08959278643280350875652641454, −0.62732580722714540881054367639,
0.49735540825034038030792428674, 1.87587789138386675822144014161, 2.66311887735582819427979752621, 3.21708378704450306184138164845, 4.40055254362032594129800426425, 4.99631518206245045567672705998, 5.62431570813914194430219641439, 6.52742408481212551525864748294, 7.43778613584404990610063263761, 8.078047998363631003150642298358
