L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s − 13-s − 14-s + 16-s + 6·17-s − 18-s + 21-s + 22-s − 24-s + 26-s + 27-s + 28-s + 6·29-s + 31-s − 32-s − 33-s − 6·34-s + 36-s − 5·37-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 − 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.218·21-s + 0.213·22-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.179·31-s − 0.176·32-s − 0.174·33-s − 1.02·34-s + 1/6·36-s − 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.879006358\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.879006358\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.171212384042543086064243682088, −7.83184202406839612692424327398, −7.12294714368617492224790851622, −6.27793805699053861891692550187, −5.39518024413601363623191450949, −4.61249338545131996728685597096, −3.51740322183097518164299590574, −2.80318452248472544711549951849, −1.85305167374744620205957579310, −0.846558027478012916629995258008,
0.846558027478012916629995258008, 1.85305167374744620205957579310, 2.80318452248472544711549951849, 3.51740322183097518164299590574, 4.61249338545131996728685597096, 5.39518024413601363623191450949, 6.27793805699053861891692550187, 7.12294714368617492224790851622, 7.83184202406839612692424327398, 8.171212384042543086064243682088