L(s) = 1 | − 1.41i·5-s + 2·7-s + 4.24i·11-s + 2.82i·13-s + 7.07i·17-s + (−1 − 4.24i)19-s + 1.41i·23-s + 2.99·25-s − 10·29-s + 2.82i·31-s − 2.82i·35-s + 5.65i·37-s + 10·41-s + 12·43-s − 1.41i·47-s + ⋯ |
L(s) = 1 | − 0.632i·5-s + 0.755·7-s + 1.27i·11-s + 0.784i·13-s + 1.71i·17-s + (−0.229 − 0.973i)19-s + 0.294i·23-s + 0.599·25-s − 1.85·29-s + 0.508i·31-s − 0.478i·35-s + 0.929i·37-s + 1.56·41-s + 1.82·43-s − 0.206i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.675006653\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.675006653\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 4.24i)T \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 - 2.82iT - 31T^{2} \) |
| 37 | \( 1 - 5.65iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 11.3iT - 79T^{2} \) |
| 83 | \( 1 - 4.24iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 8.48iT - 97T^{2} \) |
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
−9.442187623633944833672928784863, −9.028401097517210982921788062384, −8.075771189874845261506942247365, −7.36055389012067583927051587305, −6.45817429480847979334115785106, −5.38833416481285594707329183524, −4.56176659526076967099782153703, −3.94977891689844462860883213180, −2.25532421018789825959488073162, −1.40742200417023422299684696570,
0.73690625139353493193060173269, 2.37413099458576657242682178528, 3.26081149359672097723831814931, 4.30487931468662572123154714922, 5.53403863207904805216335855968, 5.94722235434439467794277978307, 7.31226398822311594873483251285, 7.68440395351289788178872173461, 8.700336458057566366144449959824, 9.402570657259317195297189230545