| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s + 13-s + 16-s − 8·19-s + 6·22-s − 26-s + 6·29-s − 2·31-s − 32-s − 2·37-s + 8·38-s + 10·43-s − 6·44-s + 12·47-s + 52-s + 12·53-s − 6·58-s − 2·61-s + 2·62-s + 64-s + 4·67-s + 2·73-s + 2·74-s − 8·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 0.277·13-s + 1/4·16-s − 1.83·19-s + 1.27·22-s − 0.196·26-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.328·37-s + 1.29·38-s + 1.52·43-s − 0.904·44-s + 1.75·47-s + 0.138·52-s + 1.64·53-s − 0.787·58-s − 0.256·61-s + 0.254·62-s + 1/8·64-s + 0.488·67-s + 0.234·73-s + 0.232·74-s − 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484739241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484739241\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.70851866921304, −12.31646606558160, −11.81174604744725, −11.11766512196427, −10.68780456657697, −10.48246585439956, −10.19164467320505, −9.472530043770771, −8.851856761803996, −8.650577297911418, −8.133201152100354, −7.600855256546017, −7.344454613407834, −6.599413522573106, −6.234545300179147, −5.630940267428286, −5.241141121052207, −4.550517347214594, −4.091845162663986, −3.417301488899396, −2.683965985558221, −2.318410503981146, −1.936848961640638, −0.8320267935775542, −0.4683057272934784,
0.4683057272934784, 0.8320267935775542, 1.936848961640638, 2.318410503981146, 2.683965985558221, 3.417301488899396, 4.091845162663986, 4.550517347214594, 5.241141121052207, 5.630940267428286, 6.234545300179147, 6.599413522573106, 7.344454613407834, 7.600855256546017, 8.133201152100354, 8.650577297911418, 8.851856761803996, 9.472530043770771, 10.19164467320505, 10.48246585439956, 10.68780456657697, 11.11766512196427, 11.81174604744725, 12.31646606558160, 12.70851866921304
