| L(s) = 1 | + 81·3-s − 8.57e3·7-s + 6.56e3·9-s + 7.05e4·11-s + 2.53e3·13-s + 2.00e5·17-s − 6.95e5·19-s − 6.94e5·21-s − 2.47e6·23-s + 5.31e5·27-s + 5.47e6·29-s + 3.73e6·31-s + 5.71e6·33-s + 2.18e7·37-s + 2.04e5·39-s − 2.38e7·41-s − 1.06e7·43-s − 2.39e6·47-s + 3.31e7·49-s + 1.62e7·51-s + 8.99e6·53-s − 5.63e7·57-s − 1.43e8·59-s − 1.98e7·61-s − 5.62e7·63-s + 1.65e8·67-s − 2.00e8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.35·7-s + 1/3·9-s + 1.45·11-s + 0.0245·13-s + 0.582·17-s − 1.22·19-s − 0.779·21-s − 1.84·23-s + 0.192·27-s + 1.43·29-s + 0.725·31-s + 0.839·33-s + 1.92·37-s + 0.0141·39-s − 1.31·41-s − 0.473·43-s − 0.0716·47-s + 0.822·49-s + 0.336·51-s + 0.156·53-s − 0.707·57-s − 1.54·59-s − 0.183·61-s − 0.450·63-s + 1.00·67-s − 1.06·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{4} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8576 T + p^{9} T^{2} \) |
| 11 | \( 1 - 70596 T + p^{9} T^{2} \) |
| 13 | \( 1 - 2530 T + p^{9} T^{2} \) |
| 17 | \( 1 - 200574 T + p^{9} T^{2} \) |
| 19 | \( 1 + 695620 T + p^{9} T^{2} \) |
| 23 | \( 1 + 2472696 T + p^{9} T^{2} \) |
| 29 | \( 1 - 188766 p T + p^{9} T^{2} \) |
| 31 | \( 1 - 3732104 T + p^{9} T^{2} \) |
| 37 | \( 1 - 21898522 T + p^{9} T^{2} \) |
| 41 | \( 1 + 580950 p T + p^{9} T^{2} \) |
| 43 | \( 1 + 10612676 T + p^{9} T^{2} \) |
| 47 | \( 1 + 2398464 T + p^{9} T^{2} \) |
| 53 | \( 1 - 8994978 T + p^{9} T^{2} \) |
| 59 | \( 1 + 143417916 T + p^{9} T^{2} \) |
| 61 | \( 1 + 19804258 T + p^{9} T^{2} \) |
| 67 | \( 1 - 165625156 T + p^{9} T^{2} \) |
| 71 | \( 1 + 194801400 T + p^{9} T^{2} \) |
| 73 | \( 1 + 148729418 T + p^{9} T^{2} \) |
| 79 | \( 1 + 30134152 T + p^{9} T^{2} \) |
| 83 | \( 1 + 302054076 T + p^{9} T^{2} \) |
| 89 | \( 1 - 909502650 T + p^{9} T^{2} \) |
| 97 | \( 1 - 872463358 T + p^{9} T^{2} \) |
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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
−9.737433061138660473782417109175, −8.826818638312926886667906088971, −7.920934113212199623612316755677, −6.51469219201828407245168819324, −6.24329731752849741634672475930, −4.38376272848776228299132507846, −3.60481626579606381114743084459, −2.57218916509312033896265561013, −1.29774629103632521095208628542, 0,
1.29774629103632521095208628542, 2.57218916509312033896265561013, 3.60481626579606381114743084459, 4.38376272848776228299132507846, 6.24329731752849741634672475930, 6.51469219201828407245168819324, 7.920934113212199623612316755677, 8.826818638312926886667906088971, 9.737433061138660473782417109175
