L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s + 6·11-s − 12-s + 13-s − 2·14-s + 16-s − 3·17-s + 18-s + 8·19-s + 2·21-s + 6·22-s − 6·23-s − 24-s + 26-s − 27-s − 2·28-s + 6·29-s − 31-s + 32-s − 6·33-s − 3·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.83·19-s + 0.436·21-s + 1.27·22-s − 1.25·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.179·31-s + 0.176·32-s − 1.04·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.750872246\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.750872246\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.030203343845990145542308337238, −7.02457460066935554440377567696, −6.63812012137035172479884132064, −6.04318018485275291213215183850, −5.33529325822239490746731965583, −4.39822919473578673661399149848, −3.78500634623640879095991548840, −3.09398561921022496550153090357, −1.84350450400433484966259689361, −0.853394087066945389387686469717,
0.853394087066945389387686469717, 1.84350450400433484966259689361, 3.09398561921022496550153090357, 3.78500634623640879095991548840, 4.39822919473578673661399149848, 5.33529325822239490746731965583, 6.04318018485275291213215183850, 6.63812012137035172479884132064, 7.02457460066935554440377567696, 8.030203343845990145542308337238