| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 7-s + 8-s + 9-s − 2·10-s − 12-s + 6·13-s + 14-s + 2·15-s + 16-s − 17-s + 18-s − 19-s − 2·20-s − 21-s − 8·23-s − 24-s − 25-s + 6·26-s − 27-s + 28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.229·19-s − 0.447·20-s − 0.218·21-s − 1.66·23-s − 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.21261537436746, −15.90284249313282, −15.51850310534038, −14.80195475360249, −14.21272665298899, −13.68817545006649, −12.92698106954521, −12.68566940389491, −11.65720429374522, −11.49300966601184, −11.19922167794424, −10.24384494008712, −9.896834959631980, −8.685834172434827, −8.250254149425815, −7.756020106677459, −6.907512902541461, −6.238035965484654, −5.897274382756004, −4.978968676312469, −4.324290915119967, −3.847438931713403, −3.195222694046241, −2.025217528239841, −1.226214823793492, 0,
1.226214823793492, 2.025217528239841, 3.195222694046241, 3.847438931713403, 4.324290915119967, 4.978968676312469, 5.897274382756004, 6.238035965484654, 6.907512902541461, 7.756020106677459, 8.250254149425815, 8.685834172434827, 9.896834959631980, 10.24384494008712, 11.19922167794424, 11.49300966601184, 11.65720429374522, 12.68566940389491, 12.92698106954521, 13.68817545006649, 14.21272665298899, 14.80195475360249, 15.51850310534038, 15.90284249313282, 16.21261537436746
