| L(s) = 1 | + 6.69i·3-s + 2.80·5-s − 75.3·7-s + 36.1·9-s + 115.·11-s + 165. i·13-s + 18.7i·15-s + 160.·17-s + (−262. + 247. i)19-s − 504. i·21-s − 1.01e3·23-s − 617.·25-s + 784. i·27-s − 1.58e3i·29-s − 439. i·31-s + ⋯ |
| L(s) = 1 | + 0.743i·3-s + 0.112·5-s − 1.53·7-s + 0.446·9-s + 0.953·11-s + 0.979i·13-s + 0.0835i·15-s + 0.555·17-s + (−0.727 + 0.686i)19-s − 1.14i·21-s − 1.91·23-s − 0.987·25-s + 1.07i·27-s − 1.88i·29-s − 0.457i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.07704204893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07704204893\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (262. - 247. i)T \) |
| good | 3 | \( 1 - 6.69iT - 81T^{2} \) |
| 5 | \( 1 - 2.80T + 625T^{2} \) |
| 7 | \( 1 + 75.3T + 2.40e3T^{2} \) |
| 11 | \( 1 - 115.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 165. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 160.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 1.01e3T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.58e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 439. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 664. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 3.20e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.09e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.41e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.51e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.88e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.55e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.63e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 9.24e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 6.35e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.01e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 5.48e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 844. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.13e4iT - 8.85e7T^{2} \) |
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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
−11.79556472864497456717192951745, −10.38549288208699034728828665677, −9.709968428842648705813408370141, −9.296787936890098038761904361185, −7.85284950244221292097427587691, −6.53306185799755428905678405125, −5.96054153135230293657572105817, −4.12960145805220737262602751734, −3.76328760436080276918267633103, −1.97060384334162262561956965382,
0.02319773885020349639939474833, 1.44380290178304948632532470303, 2.97121642386778742605328637202, 4.09573873383292498724929877945, 5.84926543673479374722833470224, 6.55325293358066098217286266119, 7.40414153262713690787135062808, 8.557103963361601197668842215637, 9.737350281571618622679013410397, 10.22689922858117691568377153130
