| L(s) = 1 | − 3.46i·5-s − 3.46i·11-s + (−1 + 3.46i)13-s − 6·17-s − 6.92i·19-s − 6.99·25-s + 6·29-s − 6.92i·31-s + 3.46i·41-s + 8·43-s − 3.46i·47-s + 7·49-s − 6·53-s − 11.9·55-s + 3.46i·59-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s − 1.04i·11-s + (−0.277 + 0.960i)13-s − 1.45·17-s − 1.58i·19-s − 1.39·25-s + 1.11·29-s − 1.24i·31-s + 0.541i·41-s + 1.21·43-s − 0.505i·47-s + 49-s − 0.824·53-s − 1.61·55-s + 0.450i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.701038 - 0.932036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.701038 - 0.932036i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 3.46iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 17.3iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−11.00450209559969519563621392240, −9.560541006232417107701333148752, −8.896891820352461166039099221199, −8.398184949689192623704007256816, −7.05430626857289287064521724670, −6.00350606340700389568574048360, −4.83635338563731504861949597010, −4.21364822268135845997944136866, −2.40580320363068386983213260792, −0.71269418439072978001669117806,
2.16207134825476194160796893893, 3.21911929507816751183128902257, 4.45479502757391604951154689181, 5.83952147512136540019032635162, 6.79858961152746685602286933351, 7.44526207867784272423902198382, 8.496957817024368779771101955302, 9.840982918322004636469544073841, 10.40836296940295937519389680938, 11.06186706002583506648137421588
