| L(s) = 1 | + 3-s + 9-s − 2·11-s − 4·13-s + 17-s + 23-s + 27-s − 2·29-s − 5·31-s − 2·33-s − 4·37-s − 4·39-s + 9·41-s + 4·43-s + 11·47-s + 51-s − 6·59-s + 6·61-s + 69-s − 3·71-s + 6·73-s + 15·79-s + 81-s − 6·83-s − 2·87-s − 9·89-s − 5·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.242·17-s + 0.208·23-s + 0.192·27-s − 0.371·29-s − 0.898·31-s − 0.348·33-s − 0.657·37-s − 0.640·39-s + 1.40·41-s + 0.609·43-s + 1.60·47-s + 0.140·51-s − 0.781·59-s + 0.768·61-s + 0.120·69-s − 0.356·71-s + 0.702·73-s + 1.68·79-s + 1/9·81-s − 0.658·83-s − 0.214·87-s − 0.953·89-s − 0.518·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.58653328953517, −14.00878952252375, −13.77295885419125, −12.99256950920324, −12.50525866197201, −12.34541091641434, −11.53348090642112, −10.87792824846954, −10.57094897623618, −9.868668444740323, −9.434194421506466, −8.993221385487099, −8.376941054100666, −7.662941425196875, −7.462534960812488, −6.905093646405913, −6.118738631950364, −5.424464037238975, −5.086002958748732, −4.242003229647009, −3.817130183080438, −2.951594185775994, −2.502684785588920, −1.904204835475993, −0.9490625489376615, 0,
0.9490625489376615, 1.904204835475993, 2.502684785588920, 2.951594185775994, 3.817130183080438, 4.242003229647009, 5.086002958748732, 5.424464037238975, 6.118738631950364, 6.905093646405913, 7.462534960812488, 7.662941425196875, 8.376941054100666, 8.993221385487099, 9.434194421506466, 9.868668444740323, 10.57094897623618, 10.87792824846954, 11.53348090642112, 12.34541091641434, 12.50525866197201, 12.99256950920324, 13.77295885419125, 14.00878952252375, 14.58653328953517
