| L(s) = 1 | + 0.864i·7-s − 3.90·11-s − 1.13·13-s − 3.71i·17-s + 1.72i·19-s + 9.03·23-s + 1.26i·29-s + 3.25i·31-s − 6.38·37-s − 6.39i·41-s − 4.77i·43-s − 4.59·47-s + 6.25·49-s + 8.98i·53-s + 8.50·59-s + ⋯ |
| L(s) = 1 | + 0.326i·7-s − 1.17·11-s − 0.314·13-s − 0.900i·17-s + 0.396i·19-s + 1.88·23-s + 0.235i·29-s + 0.584i·31-s − 1.05·37-s − 0.998i·41-s − 0.728i·43-s − 0.670·47-s + 0.893·49-s + 1.23i·53-s + 1.10·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.093863007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093863007\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 0.864iT - 7T^{2} \) |
| 11 | \( 1 + 3.90T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 + 3.71iT - 17T^{2} \) |
| 19 | \( 1 - 1.72iT - 19T^{2} \) |
| 23 | \( 1 - 9.03T + 23T^{2} \) |
| 29 | \( 1 - 1.26iT - 29T^{2} \) |
| 31 | \( 1 - 3.25iT - 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 + 6.39iT - 41T^{2} \) |
| 43 | \( 1 + 4.77iT - 43T^{2} \) |
| 47 | \( 1 + 4.59T + 47T^{2} \) |
| 53 | \( 1 - 8.98iT - 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 - 11.0iT - 67T^{2} \) |
| 71 | \( 1 - 8.10T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 + 14.2iT - 79T^{2} \) |
| 83 | \( 1 - 8.10T + 83T^{2} \) |
| 89 | \( 1 - 3.56iT - 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.091731085056496523874799086991, −7.26725571452089843310809862264, −6.99020156849707454497697531438, −5.88806425951177205944033185867, −5.14911529443120394690193669108, −4.87947289751634454955933479850, −3.65060307243604074730522312626, −2.88012160000052753337808051830, −2.23177294681288996741266411928, −0.968147040408770659849810108576,
0.30011146232006298849691846164, 1.50953266960931074906199370426, 2.57125633078204057177721757761, 3.22749384328216555661462050537, 4.18180243882096917340444277450, 5.00120958073113023311031622383, 5.44825106470789688381856287627, 6.47535387844405639238829532555, 7.00582670933271339580866699229, 7.80922174565883847952931769566
