| L(s) = 1 | − 4·7-s − 11-s + 6·13-s + 6·17-s + 8·19-s − 6·29-s + 6·37-s + 10·41-s − 8·43-s + 9·49-s − 6·53-s + 4·59-s + 2·61-s − 12·67-s + 8·71-s − 2·73-s + 4·77-s − 4·79-s + 12·83-s + 6·89-s − 24·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.301·11-s + 1.66·13-s + 1.45·17-s + 1.83·19-s − 1.11·29-s + 0.986·37-s + 1.56·41-s − 1.21·43-s + 9/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s − 1.46·67-s + 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.450·79-s + 1.31·83-s + 0.635·89-s − 2.51·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.707162791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.707162791\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.19390871611116, −12.97099471742976, −12.40402430360007, −11.82400830389943, −11.49312444643168, −10.81627847839433, −10.47209020103057, −9.734954418247865, −9.522972378891072, −9.212888387240379, −8.409155834187018, −7.922953892969535, −7.462660391356838, −6.977458808353743, −6.264387058918587, −5.915580998852258, −5.566718066917480, −4.940136259219671, −4.011044267625518, −3.600242902987254, −3.156291784643395, −2.815008511582546, −1.762156271508625, −1.063899614468736, −0.5637857957195570,
0.5637857957195570, 1.063899614468736, 1.762156271508625, 2.815008511582546, 3.156291784643395, 3.600242902987254, 4.011044267625518, 4.940136259219671, 5.566718066917480, 5.915580998852258, 6.264387058918587, 6.977458808353743, 7.462660391356838, 7.922953892969535, 8.409155834187018, 9.212888387240379, 9.522972378891072, 9.734954418247865, 10.47209020103057, 10.81627847839433, 11.49312444643168, 11.82400830389943, 12.40402430360007, 12.97099471742976, 13.19390871611116
