| L(s) = 1 | + 2·3-s − 4·7-s + 9-s − 6·11-s − 2·13-s − 4·19-s − 8·21-s − 5·25-s − 4·27-s − 4·31-s − 12·33-s − 4·37-s − 4·39-s − 6·41-s + 8·43-s + 9·49-s + 6·53-s − 8·57-s − 4·61-s − 4·63-s + 8·67-s − 2·73-s − 10·75-s + 24·77-s + 8·79-s − 11·81-s − 6·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.917·19-s − 1.74·21-s − 25-s − 0.769·27-s − 0.718·31-s − 2.08·33-s − 0.657·37-s − 0.640·39-s − 0.937·41-s + 1.21·43-s + 9/7·49-s + 0.824·53-s − 1.05·57-s − 0.512·61-s − 0.503·63-s + 0.977·67-s − 0.234·73-s − 1.15·75-s + 2.73·77-s + 0.900·79-s − 1.22·81-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6394067696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6394067696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.74775914425607, −15.12318734667989, −14.89915926555749, −13.82659814222981, −13.72320711619877, −13.03993311679758, −12.66889927609769, −12.18999116435409, −11.21059271079593, −10.54185244713209, −10.08092068536098, −9.564119133765275, −9.050010623630776, −8.353268087415553, −7.903213785085241, −7.262613553954474, −6.719595290572091, −5.789360901577994, −5.427257221399841, −4.401732064826470, −3.652854559245467, −3.092731273492740, −2.479907931247419, −2.015532955645385, −0.2821296255406784,
0.2821296255406784, 2.015532955645385, 2.479907931247419, 3.092731273492740, 3.652854559245467, 4.401732064826470, 5.427257221399841, 5.789360901577994, 6.719595290572091, 7.262613553954474, 7.903213785085241, 8.353268087415553, 9.050010623630776, 9.564119133765275, 10.08092068536098, 10.54185244713209, 11.21059271079593, 12.18999116435409, 12.66889927609769, 13.03993311679758, 13.72320711619877, 13.82659814222981, 14.89915926555749, 15.12318734667989, 15.74775914425607
