| L(s) = 1 | − 2.62i·5-s + 0.815i·7-s + 2.98·11-s − 1.44·13-s + 8.05i·17-s − i·19-s + 2.16·23-s − 1.89·25-s + 9.88i·29-s + 10.1i·31-s + 2.14·35-s − 7.24·37-s − 3.87i·41-s − 11.2i·43-s + 12.5·47-s + ⋯ |
| L(s) = 1 | − 1.17i·5-s + 0.308i·7-s + 0.901·11-s − 0.400·13-s + 1.95i·17-s − 0.229i·19-s + 0.450·23-s − 0.379·25-s + 1.83i·29-s + 1.83i·31-s + 0.361·35-s − 1.19·37-s − 0.605i·41-s − 1.71i·43-s + 1.83·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.761607002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.761607002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.62iT - 5T^{2} \) |
| 7 | \( 1 - 0.815iT - 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 + 1.44T + 13T^{2} \) |
| 17 | \( 1 - 8.05iT - 17T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 - 9.88iT - 29T^{2} \) |
| 31 | \( 1 - 10.1iT - 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 + 3.87iT - 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 8.04iT - 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 - 4.22T + 61T^{2} \) |
| 67 | \( 1 + 11.4iT - 67T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 15.4iT - 79T^{2} \) |
| 83 | \( 1 + 0.745T + 83T^{2} \) |
| 89 | \( 1 - 6.82iT - 89T^{2} \) |
| 97 | \( 1 + 0.975T + 97T^{2} \) |
| 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
−8.748115123058955538541531974701, −8.547159096721573871906048397252, −7.28752733013918097822690360153, −6.65045331655931301860198415939, −5.60352934988165922982185719671, −5.07736422048519314445641064763, −4.13320029812814910894986837816, −3.37149739611038553537338944255, −1.90696885454559848550403061806, −1.11338478379528762336503230582,
0.66175930849464755318346551190, 2.26753154259086900556960266731, 2.97039114771231468381288853576, 3.95403976756810728825249602107, 4.75207016491370065788418646695, 5.88216372759936818820745068430, 6.55605846702199829570508453165, 7.32102872171368568314935882781, 7.68899040643597537685446063607, 8.914142638524179608758087810989
