| L(s) = 1 | − 7-s + 6·11-s + 4·13-s − 6·17-s − 5·19-s + 23-s − 5·25-s + 29-s + 8·31-s − 8·37-s + 7·41-s − 10·43-s + 5·47-s + 49-s + 11·53-s + 2·61-s + 9·67-s + 15·71-s + 3·73-s − 6·77-s − 12·79-s − 6·83-s − 11·89-s − 4·91-s − 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s + 1.10·13-s − 1.45·17-s − 1.14·19-s + 0.208·23-s − 25-s + 0.185·29-s + 1.43·31-s − 1.31·37-s + 1.09·41-s − 1.52·43-s + 0.729·47-s + 1/7·49-s + 1.51·53-s + 0.256·61-s + 1.09·67-s + 1.78·71-s + 0.351·73-s − 0.683·77-s − 1.35·79-s − 0.658·83-s − 1.16·89-s − 0.419·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.165016373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.165016373\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.98732971544022, −15.40633277803545, −15.25561135039871, −14.21173299314803, −13.96266254328649, −13.30269906689038, −12.82417783167571, −12.04077492754122, −11.59573337679629, −11.07794362320327, −10.43915871229542, −9.752438494529311, −9.135144668257918, −8.542831486545680, −8.325738019612555, −7.039320937297808, −6.648679091091171, −6.268948164846522, −5.528109451682464, −4.388433180894877, −4.082364725678433, −3.436769158868185, −2.361576628543946, −1.636197574591298, −0.6601504230912159,
0.6601504230912159, 1.636197574591298, 2.361576628543946, 3.436769158868185, 4.082364725678433, 4.388433180894877, 5.528109451682464, 6.268948164846522, 6.648679091091171, 7.039320937297808, 8.325738019612555, 8.542831486545680, 9.135144668257918, 9.752438494529311, 10.43915871229542, 11.07794362320327, 11.59573337679629, 12.04077492754122, 12.82417783167571, 13.30269906689038, 13.96266254328649, 14.21173299314803, 15.25561135039871, 15.40633277803545, 15.98732971544022
