| L(s) = 1 | + 2·5-s + 2·7-s − 13-s − 17-s + 7·19-s + 2·23-s − 25-s + 2·29-s + 2·31-s + 4·35-s − 8·37-s + 10·41-s + 5·43-s + 47-s − 3·49-s − 6·53-s − 12·59-s − 2·65-s − 9·67-s − 6·71-s + 14·73-s + 16·79-s − 17·83-s − 2·85-s + 15·89-s − 2·91-s + 14·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 0.277·13-s − 0.242·17-s + 1.60·19-s + 0.417·23-s − 1/5·25-s + 0.371·29-s + 0.359·31-s + 0.676·35-s − 1.31·37-s + 1.56·41-s + 0.762·43-s + 0.145·47-s − 3/7·49-s − 0.824·53-s − 1.56·59-s − 0.248·65-s − 1.09·67-s − 0.712·71-s + 1.63·73-s + 1.80·79-s − 1.86·83-s − 0.216·85-s + 1.58·89-s − 0.209·91-s + 1.43·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.88092351014047, −12.45316985710599, −12.04744591961742, −11.52535035308405, −11.06222882368435, −10.70486435717310, −10.10111983950746, −9.738152815128317, −9.154915250425924, −9.026646340576021, −8.257598763541319, −7.688359645392537, −7.497081778203227, −6.865418046143957, −6.184656608137036, −5.936317745875794, −5.238598861255164, −4.942493375377811, −4.484289802917388, −3.720858003099588, −3.159899238938307, −2.578034976753902, −2.068255415875318, −1.380331596759057, −1.025127610447222, 0,
1.025127610447222, 1.380331596759057, 2.068255415875318, 2.578034976753902, 3.159899238938307, 3.720858003099588, 4.484289802917388, 4.942493375377811, 5.238598861255164, 5.936317745875794, 6.184656608137036, 6.865418046143957, 7.497081778203227, 7.688359645392537, 8.257598763541319, 9.026646340576021, 9.154915250425924, 9.738152815128317, 10.10111983950746, 10.70486435717310, 11.06222882368435, 11.52535035308405, 12.04744591961742, 12.45316985710599, 12.88092351014047
