| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 2·11-s − 2·13-s + 15-s − 7·17-s − 6·19-s − 3·21-s + 23-s + 25-s + 27-s − 9·29-s + 9·31-s − 2·33-s − 3·35-s − 7·37-s − 2·39-s + 5·41-s + 45-s + 8·47-s + 2·49-s − 7·51-s − 11·53-s − 2·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.258·15-s − 1.69·17-s − 1.37·19-s − 0.654·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.67·29-s + 1.61·31-s − 0.348·33-s − 0.507·35-s − 1.15·37-s − 0.320·39-s + 0.780·41-s + 0.149·45-s + 1.16·47-s + 2/7·49-s − 0.980·51-s − 1.51·53-s − 0.269·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.114626272342420246330424630098, −8.608402893779175259110116582323, −7.50086256368661123062194361771, −6.69307252699898801657576735237, −6.05722317603077213338585769719, −4.84828956114251062555624822682, −3.94385539988496915567759022760, −2.77431282556833018538797503915, −2.09630997874818263004859828457, 0,
2.09630997874818263004859828457, 2.77431282556833018538797503915, 3.94385539988496915567759022760, 4.84828956114251062555624822682, 6.05722317603077213338585769719, 6.69307252699898801657576735237, 7.50086256368661123062194361771, 8.608402893779175259110116582323, 9.114626272342420246330424630098
