| L(s) = 1 | − 2-s + 3-s + 4-s + 3.29·5-s − 6-s − 8-s + 9-s − 3.29·10-s − 11-s + 12-s − 3.23·13-s + 3.29·15-s + 16-s + 0.462·17-s − 18-s + 2.77·19-s + 3.29·20-s + 22-s − 2.90·23-s − 24-s + 5.82·25-s + 3.23·26-s + 27-s + 2.82·29-s − 3.29·30-s − 0.704·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.47·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.04·10-s − 0.301·11-s + 0.288·12-s − 0.898·13-s + 0.849·15-s + 0.250·16-s + 0.112·17-s − 0.235·18-s + 0.637·19-s + 0.735·20-s + 0.213·22-s − 0.604·23-s − 0.204·24-s + 1.16·25-s + 0.635·26-s + 0.192·27-s + 0.525·29-s − 0.600·30-s − 0.126·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.202740981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.202740981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.29T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 0.462T + 17T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 0.704T + 31T^{2} \) |
| 37 | \( 1 - 0.996T + 37T^{2} \) |
| 41 | \( 1 + 0.462T + 41T^{2} \) |
| 43 | \( 1 - 6.75T + 43T^{2} \) |
| 47 | \( 1 - 3.94T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 2.06T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 0.359T + 73T^{2} \) |
| 79 | \( 1 - 0.168T + 79T^{2} \) |
| 83 | \( 1 - 7.53T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 3.89T + 97T^{2} \) |
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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
−8.813216580689472940135712156114, −7.982605087774656844948992136370, −7.27405874261582228956675615810, −6.53798974431846765200614055306, −5.66751349203700436250226642909, −5.04846149200545799243478761073, −3.77466704932549821531615731445, −2.52639349065783285134585201865, −2.22237085752925003982935036300, −0.987418545566388354409682701717,
0.987418545566388354409682701717, 2.22237085752925003982935036300, 2.52639349065783285134585201865, 3.77466704932549821531615731445, 5.04846149200545799243478761073, 5.66751349203700436250226642909, 6.53798974431846765200614055306, 7.27405874261582228956675615810, 7.982605087774656844948992136370, 8.813216580689472940135712156114
