L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 2·7-s + 3·8-s + 9-s + 10-s − 12-s + 2·14-s − 15-s − 16-s − 2·17-s − 18-s − 2·19-s + 20-s − 2·21-s + 8·23-s + 3·24-s + 25-s + 27-s + 2·28-s + 2·29-s + 30-s + 2·31-s − 5·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.436·21-s + 1.66·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.371·29-s + 0.182·30-s + 0.359·31-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.575953348107293485763937975608, −8.021594126447427734832418385345, −7.10540557581421844568439597991, −6.56805494128641193611634338112, −5.22306552180917279494507366447, −4.44318600575075124216921923979, −3.59257176671015508512285767256, −2.70186621341530405399067473821, −1.31733879076898618409918721655, 0,
1.31733879076898618409918721655, 2.70186621341530405399067473821, 3.59257176671015508512285767256, 4.44318600575075124216921923979, 5.22306552180917279494507366447, 6.56805494128641193611634338112, 7.10540557581421844568439597991, 8.021594126447427734832418385345, 8.575953348107293485763937975608