| L(s) = 1 | − 3.46i·5-s + 2.64·7-s − 4.58i·13-s − 1.73i·17-s + 5.29·19-s − 4.58i·23-s − 6.99·25-s − 7.93·29-s + 2.64·31-s − 9.16i·35-s + 4·37-s + 6.92i·41-s − 1.73i·43-s − 6·47-s + 7.00·49-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s + 0.999·7-s − 1.27i·13-s − 0.420i·17-s + 1.21·19-s − 0.955i·23-s − 1.39·25-s − 1.47·29-s + 0.475·31-s − 1.54i·35-s + 0.657·37-s + 1.08i·41-s − 0.264i·43-s − 0.875·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944191279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944191279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.64T \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4.58iT - 13T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 4.58iT - 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 9.16iT - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 4.58iT - 71T^{2} \) |
| 73 | \( 1 - 9.16iT - 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 5.19iT - 89T^{2} \) |
| 97 | \( 1 - 9.16iT - 97T^{2} \) |
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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
−8.232733799308681442854838464463, −8.064330762535243250199747163934, −7.18723677089450082100859119608, −5.89345151799220228361544062480, −5.22563979237517673582218421057, −4.81265479800648420960377046923, −3.88018829935572687095994075020, −2.68168200761265926274422247207, −1.44282694542579214612777598933, −0.64137665604591634232310830250,
1.53900610636398936446682359892, 2.36889557002579166584904714467, 3.43075461060917871380850270448, 4.12535876584937275948195842457, 5.21518871867730154009282005862, 5.98981763710270761208689160428, 6.86738925948968322891126338515, 7.42264508016960146909166815705, 7.982416552004639699866978861730, 9.102516645490947010124479324550
