| L(s) = 1 | + 7.23·5-s − 93.1·7-s + 169.·11-s + 534.·17-s − 361·19-s − 1.04e3·23-s − 572.·25-s − 673.·35-s − 800.·43-s + 3.17e3·47-s + 6.27e3·49-s + 1.22e3·55-s + 7.41e3·61-s + 1.90e3·73-s − 1.57e4·77-s + 1.25e4·83-s + 3.86e3·85-s − 2.61e3·95-s + 1.77e4·101-s − 7.56e3·115-s − 4.97e4·119-s + ⋯ |
| L(s) = 1 | + 0.289·5-s − 1.90·7-s + 1.39·11-s + 1.84·17-s − 19-s − 1.97·23-s − 0.916·25-s − 0.549·35-s − 0.433·43-s + 1.43·47-s + 2.61·49-s + 0.404·55-s + 1.99·61-s + 0.356·73-s − 2.65·77-s + 1.82·83-s + 0.535·85-s − 0.289·95-s + 1.74·101-s − 0.572·115-s − 3.51·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.616349182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.616349182\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 361T \) |
| good | 5 | \( 1 - 7.23T + 625T^{2} \) |
| 7 | \( 1 + 93.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 169.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 - 534.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 1.04e3T + 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 + 800.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.17e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 - 7.41e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.90e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.25e4T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.85e7T^{2} \) |
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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
−9.831354033228673322601474955841, −9.313091242272222981563338649979, −8.215307077838519363522660346349, −7.08389391302658207518274607788, −6.20144199390389984280719615219, −5.79124359432111309517590110173, −3.98836464322773556253195689407, −3.48099822564587284079115598994, −2.10567316538357749232164367897, −0.64399365800456591011330930648,
0.64399365800456591011330930648, 2.10567316538357749232164367897, 3.48099822564587284079115598994, 3.98836464322773556253195689407, 5.79124359432111309517590110173, 6.20144199390389984280719615219, 7.08389391302658207518274607788, 8.215307077838519363522660346349, 9.313091242272222981563338649979, 9.831354033228673322601474955841
