| L(s) = 1 | − 4·5-s − 2·7-s − 17-s − 6·23-s + 11·25-s − 2·29-s − 4·31-s + 8·35-s + 2·37-s − 6·41-s − 4·43-s − 6·47-s − 3·49-s − 8·53-s + 8·59-s + 8·61-s + 4·67-s − 6·71-s − 10·73-s − 6·79-s + 4·83-s + 4·85-s + 14·89-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s − 0.242·17-s − 1.25·23-s + 11/5·25-s − 0.371·29-s − 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s − 1.09·53-s + 1.04·59-s + 1.02·61-s + 0.488·67-s − 0.712·71-s − 1.17·73-s − 0.675·79-s + 0.439·83-s + 0.433·85-s + 1.48·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.99167328392685, −12.81947787095293, −12.08066048109277, −11.67901157634794, −11.58914348992697, −10.94547711902342, −10.44082439831525, −9.961334014469031, −9.537137326512442, −8.854971216916289, −8.489803680833676, −8.071928525970368, −7.515429132269153, −7.263787828192149, −6.591226659768300, −6.276939909321646, −5.620787068139221, −4.851818094947785, −4.608502307834513, −3.861983297065919, −3.505648426840604, −3.265129753764886, −2.410933418696835, −1.786923543377196, −0.9159107958654783, 0, 0,
0.9159107958654783, 1.786923543377196, 2.410933418696835, 3.265129753764886, 3.505648426840604, 3.861983297065919, 4.608502307834513, 4.851818094947785, 5.620787068139221, 6.276939909321646, 6.591226659768300, 7.263787828192149, 7.515429132269153, 8.071928525970368, 8.489803680833676, 8.854971216916289, 9.537137326512442, 9.961334014469031, 10.44082439831525, 10.94547711902342, 11.58914348992697, 11.67901157634794, 12.08066048109277, 12.81947787095293, 12.99167328392685
