| L(s) = 1 | + 2-s + 4-s + 5-s − 2·7-s + 8-s + 10-s + 2·13-s − 2·14-s + 16-s + 2·17-s + 20-s + 2·23-s + 25-s + 2·26-s − 2·28-s + 4·29-s − 4·31-s + 32-s + 2·34-s − 2·35-s + 2·37-s + 40-s − 4·41-s + 10·43-s + 2·46-s − 6·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.223·20-s + 0.417·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.338·35-s + 0.328·37-s + 0.158·40-s − 0.624·41-s + 1.52·43-s + 0.294·46-s − 0.875·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.990196738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.990196738\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.92274961158028, −14.40000431937693, −14.02572756114865, −13.36414832202500, −13.01252840478297, −12.52877041326870, −12.01046749478029, −11.36853778149110, −10.73941007619347, −10.40025975369393, −9.591211347436858, −9.292683485026623, −8.551992822075644, −7.843579722691714, −7.301505608757322, −6.510248063835901, −6.234040363458558, −5.642410667082334, −4.931041198436344, −4.407067938109612, −3.388282946249632, −3.273363198819412, −2.344083273681132, −1.584174401863705, −0.6763948110231196,
0.6763948110231196, 1.584174401863705, 2.344083273681132, 3.273363198819412, 3.388282946249632, 4.407067938109612, 4.931041198436344, 5.642410667082334, 6.234040363458558, 6.510248063835901, 7.301505608757322, 7.843579722691714, 8.551992822075644, 9.292683485026623, 9.591211347436858, 10.40025975369393, 10.73941007619347, 11.36853778149110, 12.01046749478029, 12.52877041326870, 13.01252840478297, 13.36414832202500, 14.02572756114865, 14.40000431937693, 14.92274961158028
