| L(s) = 1 | + 0.319·5-s − i·7-s − 0.987·11-s − 1.19·13-s + 1.59·17-s + 7.25i·19-s + (−2.39 + 4.15i)23-s − 4.89·25-s − 0.889i·29-s + 2.30·31-s − 0.319i·35-s − 8.87i·37-s + 8.57i·41-s − 8.68i·43-s − 12.6i·47-s + ⋯ |
| L(s) = 1 | + 0.142·5-s − 0.377i·7-s − 0.297·11-s − 0.330·13-s + 0.386·17-s + 1.66i·19-s + (−0.499 + 0.866i)23-s − 0.979·25-s − 0.165i·29-s + 0.414·31-s − 0.0539i·35-s − 1.45i·37-s + 1.33i·41-s − 1.32i·43-s − 1.84i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3659245836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3659245836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 23 | \( 1 + (2.39 - 4.15i)T \) |
| good | 5 | \( 1 - 0.319T + 5T^{2} \) |
| 11 | \( 1 + 0.987T + 11T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 - 7.25iT - 19T^{2} \) |
| 29 | \( 1 + 0.889iT - 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 + 8.87iT - 37T^{2} \) |
| 41 | \( 1 - 8.57iT - 41T^{2} \) |
| 43 | \( 1 + 8.68iT - 43T^{2} \) |
| 47 | \( 1 + 12.6iT - 47T^{2} \) |
| 53 | \( 1 + 4.28T + 53T^{2} \) |
| 59 | \( 1 + 6.83iT - 59T^{2} \) |
| 61 | \( 1 - 0.558iT - 61T^{2} \) |
| 67 | \( 1 + 4.98iT - 67T^{2} \) |
| 71 | \( 1 - 0.865iT - 71T^{2} \) |
| 73 | \( 1 - 5.50T + 73T^{2} \) |
| 79 | \( 1 + 13.9iT - 79T^{2} \) |
| 83 | \( 1 + 9.61T + 83T^{2} \) |
| 89 | \( 1 + 9.00T + 89T^{2} \) |
| 97 | \( 1 + 4.55iT - 97T^{2} \) |
| 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
−7.79296096223151432668166663880, −7.26209542071680401803010748937, −6.26897847332400957202892464626, −5.68913719694855340470080670551, −5.00842511636380700487033028734, −3.91560327105376222468060188999, −3.52187692283076975494941817539, −2.27612008186325238932247371090, −1.50752589076089362617514887892, −0.092442312759410266526717403701,
1.24698753047562540759327393667, 2.48519221950451351010835481410, 2.91448255552574440397188974804, 4.15367603245748207670076888424, 4.79691080951809395474215090440, 5.54405304049783114025044078380, 6.31241396354954585111377563498, 6.94334084282809286433901208012, 7.79570227539707699476810698087, 8.344497289006543722848712326187
