L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4.53·7-s − 8-s + 9-s + 6.53·11-s − 12-s − 6·13-s − 4.53·14-s + 16-s + 4·17-s − 18-s + 4.53·19-s − 4.53·21-s − 6.53·22-s − 6.53·23-s + 24-s + 6·26-s − 27-s + 4.53·28-s + 31-s − 32-s − 6.53·33-s − 4·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.71·7-s − 0.353·8-s + 0.333·9-s + 1.96·11-s − 0.288·12-s − 1.66·13-s − 1.21·14-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 1.03·19-s − 0.988·21-s − 1.39·22-s − 1.36·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s + 0.856·28-s + 0.179·31-s − 0.176·32-s − 1.13·33-s − 0.685·34-s + 0.166·36-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
\(L(1)\) |
\(\approx\) |
\(1.669680186\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.669680186\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.53T + 7T^{2} \) |
| 11 | \( 1 - 6.53T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 23 | \( 1 + 6.53T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 - 7.06T + 37T^{2} \) |
| 41 | \( 1 - 7.06T + 41T^{2} \) |
| 43 | \( 1 - 8.53T + 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 + 5.06T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 8.53T + 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 6.53T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.153934871379927956414994173322, −7.56408231284478756861025936390, −7.18024192622616061566564159895, −6.07501867292663783385547774706, −5.51376186232524458228786901041, −4.55542202858710814296390487616, −4.01747719748240670356670143382, −2.57668487827483542201620935882, −1.58084702824352474629991976218, −0.916770209767980851110675105888,
0.916770209767980851110675105888, 1.58084702824352474629991976218, 2.57668487827483542201620935882, 4.01747719748240670356670143382, 4.55542202858710814296390487616, 5.51376186232524458228786901041, 6.07501867292663783385547774706, 7.18024192622616061566564159895, 7.56408231284478756861025936390, 8.153934871379927956414994173322