| L(s) = 1 | + (−2 − i)5-s − 4i·7-s − 4·11-s + 4i·13-s + 6i·17-s − 4·19-s + 4i·23-s + (3 + 4i)25-s − 4·29-s + (−4 + 8i)35-s − 4i·37-s − 8·41-s − 12i·47-s − 9·49-s − 2i·53-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s − 1.51i·7-s − 1.20·11-s + 1.10i·13-s + 1.45i·17-s − 0.917·19-s + 0.834i·23-s + (0.600 + 0.800i)25-s − 0.742·29-s + (−0.676 + 1.35i)35-s − 0.657i·37-s − 1.24·41-s − 1.75i·47-s − 1.28·49-s − 0.274i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−10.14648392519205838162628278275, −8.926648884687185212925284558356, −8.064326474338784466073449581357, −7.42891881658125989292916077155, −6.57419690303017447639645291802, −5.20380729829409182010252153007, −4.17257174932616932640189191169, −3.63602814170657235871932812636, −1.73120086306290826165577023814, 0,
2.52872083637152739966316508117, 3.07811712083300008315310753009, 4.68923333742045961117131458509, 5.46364590272685055679858231450, 6.48770660992820850167625129137, 7.64062808339153370273001563858, 8.226901402151902866738141356653, 9.058722085716194789880205596641, 10.15868904826550110521612182023
