| L(s) = 1 | − 3-s − 3.22·7-s + 9-s + 2·11-s + 3.22·13-s − 6.88·17-s + 4·19-s + 3.22·21-s + 6.88·23-s − 27-s − 1.65·29-s − 31-s − 2·33-s + 5.22·37-s − 3.22·39-s + 4·41-s − 2·43-s − 1.57·47-s + 3.42·49-s + 6.88·51-s − 0.425·53-s − 4·57-s + 7.65·59-s − 11.7·61-s − 3.22·63-s − 14.9·67-s − 6.88·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.22·7-s + 0.333·9-s + 0.603·11-s + 0.895·13-s − 1.66·17-s + 0.917·19-s + 0.704·21-s + 1.43·23-s − 0.192·27-s − 0.307·29-s − 0.179·31-s − 0.348·33-s + 0.859·37-s − 0.517·39-s + 0.624·41-s − 0.304·43-s − 0.229·47-s + 0.489·49-s + 0.963·51-s − 0.0584·53-s − 0.529·57-s + 0.996·59-s − 1.50·61-s − 0.406·63-s − 1.83·67-s − 0.828·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.284411177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284411177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 + 6.88T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 + 0.425T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 4.80T + 71T^{2} \) |
| 73 | \( 1 - 7.68T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.88T + 83T^{2} \) |
| 89 | \( 1 - 1.88T + 89T^{2} \) |
| 97 | \( 1 - 9.76T + 97T^{2} \) |
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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
−7.42145763350439616505974317181, −6.95436913154466363448587993494, −6.25388028951243867299564844683, −5.93729099027004314013528959097, −4.91583158359790772753341392661, −4.24012409304309623339679672132, −3.44038930099576752744971905755, −2.77106549172949314097358872545, −1.55937725495653534581038076219, −0.57882713905970231461381229355,
0.57882713905970231461381229355, 1.55937725495653534581038076219, 2.77106549172949314097358872545, 3.44038930099576752744971905755, 4.24012409304309623339679672132, 4.91583158359790772753341392661, 5.93729099027004314013528959097, 6.25388028951243867299564844683, 6.95436913154466363448587993494, 7.42145763350439616505974317181
