L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 3·11-s − 12-s − 5·13-s − 14-s + 16-s + 18-s − 4·19-s + 21-s + 3·22-s + 6·23-s − 24-s − 5·26-s − 27-s − 28-s + 5·31-s + 32-s − 3·33-s + 36-s + 4·37-s − 4·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.188·28-s + 0.898·31-s + 0.176·32-s − 0.522·33-s + 1/6·36-s + 0.657·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.87137165682308, −12.48966437054814, −12.07705478161607, −11.50350835298593, −11.29882572531160, −10.74051809291083, −10.18050249952159, −9.742465982193561, −9.414952487866323, −8.789081270813799, −8.210445500203593, −7.678076134136965, −7.036471667451064, −6.761409107001102, −6.396294138036294, −5.824343738723900, −5.274642182790355, −4.793564265778828, −4.384157988964991, −3.927209955230727, −3.235350583351543, −2.672265845111441, −2.221992264085795, −1.424435947605141, −0.8097501458678691, 0,
0.8097501458678691, 1.424435947605141, 2.221992264085795, 2.672265845111441, 3.235350583351543, 3.927209955230727, 4.384157988964991, 4.793564265778828, 5.274642182790355, 5.824343738723900, 6.396294138036294, 6.761409107001102, 7.036471667451064, 7.678076134136965, 8.210445500203593, 8.789081270813799, 9.414952487866323, 9.742465982193561, 10.18050249952159, 10.74051809291083, 11.29882572531160, 11.50350835298593, 12.07705478161607, 12.48966437054814, 12.87137165682308