L(s) = 1 | + 0.618·2-s − 3-s − 1.61·4-s − 0.618·6-s − 3·7-s − 2.23·8-s + 9-s + 1.61·12-s − 1.76·13-s − 1.85·14-s + 1.85·16-s − 1.61·17-s + 0.618·18-s + 5.85·19-s + 3·21-s − 3.47·23-s + 2.23·24-s − 1.09·26-s − 27-s + 4.85·28-s + 4.47·29-s + 2.85·31-s + 5.61·32-s − 1.00·34-s − 1.61·36-s − 0.236·37-s + 3.61·38-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.252·6-s − 1.13·7-s − 0.790·8-s + 0.333·9-s + 0.467·12-s − 0.489·13-s − 0.495·14-s + 0.463·16-s − 0.392·17-s + 0.145·18-s + 1.34·19-s + 0.654·21-s − 0.723·23-s + 0.456·24-s − 0.213·26-s − 0.192·27-s + 0.917·28-s + 0.830·29-s + 0.512·31-s + 0.993·32-s − 0.171·34-s − 0.269·36-s − 0.0388·37-s + 0.586·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 + 0.236T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 - 9.61T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 + 5.56T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 + 0.708T + 83T^{2} \) |
| 89 | \( 1 - 0.527T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.13855714806406856103502327480, −6.61276158165324829095901925691, −5.95442634374765011919170781138, −5.19215411472549953535497969974, −4.79845560539928997189654193557, −3.77039845075160798831231171483, −3.34835266650035396264172198491, −2.36130092750593575364674390660, −0.924919728282169098794694179233, 0,
0.924919728282169098794694179233, 2.36130092750593575364674390660, 3.34835266650035396264172198491, 3.77039845075160798831231171483, 4.79845560539928997189654193557, 5.19215411472549953535497969974, 5.95442634374765011919170781138, 6.61276158165324829095901925691, 7.13855714806406856103502327480