L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 5·7-s + 8-s − 2·9-s + 11-s − 12-s − 2·13-s − 5·14-s + 16-s − 3·17-s − 2·18-s − 7·19-s + 5·21-s + 22-s + 6·23-s − 24-s − 2·26-s + 5·27-s − 5·28-s − 3·29-s − 7·31-s + 32-s − 33-s − 3·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.88·7-s + 0.353·8-s − 2/3·9-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 1.33·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.60·19-s + 1.09·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 0.392·26-s + 0.962·27-s − 0.944·28-s − 0.557·29-s − 1.25·31-s + 0.176·32-s − 0.174·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−10.59296143170299319105275536153, −9.515625676781913720459205161908, −8.766840852787028026444155939270, −7.20457953717496815600620253355, −6.43810722624715010093850437821, −5.90185767008403089888905192216, −4.68830133406592764908928362464, −3.51902803001209100282256024354, −2.52121766859740692142668949704, 0,
2.52121766859740692142668949704, 3.51902803001209100282256024354, 4.68830133406592764908928362464, 5.90185767008403089888905192216, 6.43810722624715010093850437821, 7.20457953717496815600620253355, 8.766840852787028026444155939270, 9.515625676781913720459205161908, 10.59296143170299319105275536153