| L(s) = 1 | − 5-s + 2.82·7-s + 11-s − 0.828·13-s − 4.82·17-s + 5.65·19-s + 5.65·23-s + 25-s − 2·29-s − 5.65·31-s − 2.82·35-s + 6·37-s + 2·41-s + 6.82·43-s + 5.65·47-s + 1.00·49-s − 11.6·53-s − 55-s + 1.65·59-s + 11.6·61-s + 0.828·65-s − 1.65·67-s + 2.34·71-s − 2.48·73-s + 2.82·77-s − 1.65·79-s − 5.17·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.06·7-s + 0.301·11-s − 0.229·13-s − 1.17·17-s + 1.29·19-s + 1.17·23-s + 0.200·25-s − 0.371·29-s − 1.01·31-s − 0.478·35-s + 0.986·37-s + 0.312·41-s + 1.04·43-s + 0.825·47-s + 0.142·49-s − 1.60·53-s − 0.134·55-s + 0.215·59-s + 1.49·61-s + 0.102·65-s − 0.202·67-s + 0.278·71-s − 0.290·73-s + 0.322·77-s − 0.186·79-s − 0.567·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022221092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022221092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.506672084479354304379519407820, −7.51134579576658246103455937523, −7.32273447663721036735229445963, −6.26079248906831332368496824482, −5.31937873763793187716385224988, −4.70711642192822628126718332031, −3.96311334855885077276894452647, −2.95641710277325158142342473715, −1.93167828638980853879531710059, −0.843934856601641669950234890221,
0.843934856601641669950234890221, 1.93167828638980853879531710059, 2.95641710277325158142342473715, 3.96311334855885077276894452647, 4.70711642192822628126718332031, 5.31937873763793187716385224988, 6.26079248906831332368496824482, 7.32273447663721036735229445963, 7.51134579576658246103455937523, 8.506672084479354304379519407820
