| L(s) = 1 | + 5-s − 2·7-s − 2·11-s + 6·13-s − 4·17-s + 4·19-s + 25-s + 2·29-s + 4·31-s − 2·35-s − 8·37-s + 8·41-s + 43-s − 8·47-s − 3·49-s − 2·53-s − 2·55-s + 6·59-s + 4·61-s + 6·65-s − 12·67-s + 10·73-s + 4·77-s − 8·79-s − 14·83-s − 4·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.603·11-s + 1.66·13-s − 0.970·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.718·31-s − 0.338·35-s − 1.31·37-s + 1.24·41-s + 0.152·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s − 0.269·55-s + 0.781·59-s + 0.512·61-s + 0.744·65-s − 1.46·67-s + 1.17·73-s + 0.455·77-s − 0.900·79-s − 1.53·83-s − 0.433·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.59660981539388, −13.37369093862523, −12.87872117983568, −12.53818754203230, −11.72658911096037, −11.34569891135405, −10.89297793168276, −10.29892247354559, −9.971979788007265, −9.402005350750011, −8.791731853170750, −8.569756802463954, −7.902337047216667, −7.284964810701244, −6.714132108230415, −6.231567221634560, −5.921085178635964, −5.224933014938170, −4.711816119244198, −3.999451824605371, −3.405693701986251, −2.960795345219024, −2.299314822641530, −1.534266066015655, −0.9007156991671797, 0,
0.9007156991671797, 1.534266066015655, 2.299314822641530, 2.960795345219024, 3.405693701986251, 3.999451824605371, 4.711816119244198, 5.224933014938170, 5.921085178635964, 6.231567221634560, 6.714132108230415, 7.284964810701244, 7.902337047216667, 8.569756802463954, 8.791731853170750, 9.402005350750011, 9.971979788007265, 10.29892247354559, 10.89297793168276, 11.34569891135405, 11.72658911096037, 12.53818754203230, 12.87872117983568, 13.37369093862523, 13.59660981539388
