| L(s) = 1 | − 3-s + 3·5-s + 5·7-s + 9-s + 11-s − 2·13-s − 3·15-s − 17-s − 19-s − 5·21-s + 4·23-s + 4·25-s − 27-s + 2·29-s + 6·31-s − 33-s + 15·35-s + 2·39-s − 43-s + 3·45-s + 9·47-s + 18·49-s + 51-s − 10·53-s + 3·55-s + 57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1.88·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.774·15-s − 0.242·17-s − 0.229·19-s − 1.09·21-s + 0.834·23-s + 4/5·25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.174·33-s + 2.53·35-s + 0.320·39-s − 0.152·43-s + 0.447·45-s + 1.31·47-s + 18/7·49-s + 0.140·51-s − 1.37·53-s + 0.404·55-s + 0.132·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.712927601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.712927601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.562174106312293436599005698498, −7.80605317251250760198226171715, −6.97556751348491276200701022616, −6.21053798963199366539375101013, −5.43607631650156514852377977246, −4.88619376922667033041566401202, −4.27866732192520382052996066610, −2.68139800535164168175122974103, −1.84871910742097553432780734662, −1.10084134564969783818306772778,
1.10084134564969783818306772778, 1.84871910742097553432780734662, 2.68139800535164168175122974103, 4.27866732192520382052996066610, 4.88619376922667033041566401202, 5.43607631650156514852377977246, 6.21053798963199366539375101013, 6.97556751348491276200701022616, 7.80605317251250760198226171715, 8.562174106312293436599005698498
