| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 1.61·11-s − 6.31·13-s − 15-s + 7.92·17-s − 2.38·19-s + 21-s + 6.70·23-s + 25-s − 27-s + 7.92·29-s − 9.08·31-s + 1.61·33-s − 35-s − 7.92·37-s + 6.31·39-s − 1.22·41-s + 5.92·43-s + 45-s − 6.70·47-s + 49-s − 7.92·51-s − 3.61·53-s − 1.61·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s − 0.486·11-s − 1.75·13-s − 0.258·15-s + 1.92·17-s − 0.547·19-s + 0.218·21-s + 1.39·23-s + 0.200·25-s − 0.192·27-s + 1.47·29-s − 1.63·31-s + 0.280·33-s − 0.169·35-s − 1.30·37-s + 1.01·39-s − 0.191·41-s + 0.903·43-s + 0.149·45-s − 0.977·47-s + 0.142·49-s − 1.10·51-s − 0.496·53-s − 0.217·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 17 | \( 1 - 7.92T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 9.08T + 31T^{2} \) |
| 37 | \( 1 + 7.92T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 - 5.92T + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 4.70T + 61T^{2} \) |
| 67 | \( 1 + 9.14T + 67T^{2} \) |
| 71 | \( 1 + 8.31T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 0.775T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−8.153555569960952287931961357723, −7.30655399910602478594450828176, −6.88334314271073915907445009890, −5.82718950151090627940880055885, −5.23831739754880888743281094212, −4.67015979109330089027074972565, −3.34970583790651186908166138140, −2.61301686148822355848171658844, −1.37640897272277727068186361502, 0,
1.37640897272277727068186361502, 2.61301686148822355848171658844, 3.34970583790651186908166138140, 4.67015979109330089027074972565, 5.23831739754880888743281094212, 5.82718950151090627940880055885, 6.88334314271073915907445009890, 7.30655399910602478594450828176, 8.153555569960952287931961357723
