| L(s) = 1 | + 3·5-s − 7-s + 4·13-s − 4·17-s + 8·19-s + 3·23-s + 4·25-s − 4·29-s + 5·31-s − 3·35-s + 11·37-s + 4·43-s + 4·47-s + 49-s + 10·53-s − 3·59-s + 12·61-s + 12·65-s − 5·67-s + 9·71-s + 16·73-s + 8·79-s + 12·83-s − 12·85-s + 13·89-s − 4·91-s + 24·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 1.10·13-s − 0.970·17-s + 1.83·19-s + 0.625·23-s + 4/5·25-s − 0.742·29-s + 0.898·31-s − 0.507·35-s + 1.80·37-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 1.37·53-s − 0.390·59-s + 1.53·61-s + 1.48·65-s − 0.610·67-s + 1.06·71-s + 1.87·73-s + 0.900·79-s + 1.31·83-s − 1.30·85-s + 1.37·89-s − 0.419·91-s + 2.46·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.411239753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.411239753\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.00616521367803, −13.69169976496007, −13.44176488303265, −12.97958576051410, −12.36872975887050, −11.69849140074699, −11.13322173793258, −10.82751155816000, −10.06109470603841, −9.656175001673583, −9.216290387882524, −8.864965021851389, −8.066410772810019, −7.525815493660683, −6.793438295742230, −6.363637383743294, −5.871659809536131, −5.358355094747904, −4.830709312659010, −3.925157978258333, −3.460431587984771, −2.521862058810849, −2.286617958090664, −1.181481867154806, −0.8287882173918323,
0.8287882173918323, 1.181481867154806, 2.286617958090664, 2.521862058810849, 3.460431587984771, 3.925157978258333, 4.830709312659010, 5.358355094747904, 5.871659809536131, 6.363637383743294, 6.793438295742230, 7.525815493660683, 8.066410772810019, 8.864965021851389, 9.216290387882524, 9.656175001673583, 10.06109470603841, 10.82751155816000, 11.13322173793258, 11.69849140074699, 12.36872975887050, 12.97958576051410, 13.44176488303265, 13.69169976496007, 14.00616521367803
