| L(s) = 1 | − 2-s − 0.415·3-s + 4-s − 2.24·5-s + 0.415·6-s + 1.94·7-s − 8-s − 2.82·9-s + 2.24·10-s + 11-s − 0.415·12-s + 1.65·13-s − 1.94·14-s + 0.933·15-s + 16-s − 4.04·17-s + 2.82·18-s − 7.25·19-s − 2.24·20-s − 0.809·21-s − 22-s + 8.28·23-s + 0.415·24-s + 0.0380·25-s − 1.65·26-s + 2.42·27-s + 1.94·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.240·3-s + 0.5·4-s − 1.00·5-s + 0.169·6-s + 0.735·7-s − 0.353·8-s − 0.942·9-s + 0.709·10-s + 0.301·11-s − 0.120·12-s + 0.458·13-s − 0.520·14-s + 0.241·15-s + 0.250·16-s − 0.980·17-s + 0.666·18-s − 1.66·19-s − 0.501·20-s − 0.176·21-s − 0.213·22-s + 1.72·23-s + 0.0848·24-s + 0.00761·25-s − 0.324·26-s + 0.466·27-s + 0.367·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 0.415T + 3T^{2} \) |
| 5 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 8.28T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 + 0.827T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 + 7.74T + 53T^{2} \) |
| 59 | \( 1 - 7.99T + 59T^{2} \) |
| 61 | \( 1 + 8.08T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 3.83T + 71T^{2} \) |
| 73 | \( 1 - 0.425T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 2.39T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.168382996414673317154536934525, −7.46532629837641566184405371855, −6.52607053572922905393207059888, −6.13750675958258453881573181493, −4.81693163736970935315974528987, −4.39724312853568069771504701531, −3.23112003517736700744843116871, −2.39330438840473661990864761864, −1.13325173468721351225937605370, 0,
1.13325173468721351225937605370, 2.39330438840473661990864761864, 3.23112003517736700744843116871, 4.39724312853568069771504701531, 4.81693163736970935315974528987, 6.13750675958258453881573181493, 6.52607053572922905393207059888, 7.46532629837641566184405371855, 8.168382996414673317154536934525
