L(s) = 1 | + 0.568·3-s − 5-s − 4.73·7-s − 2.67·9-s + 0.360·11-s + 5.26·13-s − 0.568·15-s + 0.370·17-s + 4.60·19-s − 2.69·21-s + 23-s + 25-s − 3.22·27-s + 0.939·29-s − 9.66·31-s + 0.204·33-s + 4.73·35-s + 3.26·37-s + 2.99·39-s + 5.29·41-s + 2.67·45-s + 1.25·47-s + 15.4·49-s + 0.210·51-s + 10.9·53-s − 0.360·55-s + 2.61·57-s + ⋯ |
L(s) = 1 | + 0.328·3-s − 0.447·5-s − 1.79·7-s − 0.892·9-s + 0.108·11-s + 1.45·13-s − 0.146·15-s + 0.0899·17-s + 1.05·19-s − 0.587·21-s + 0.208·23-s + 0.200·25-s − 0.620·27-s + 0.174·29-s − 1.73·31-s + 0.0356·33-s + 0.800·35-s + 0.537·37-s + 0.478·39-s + 0.827·41-s + 0.399·45-s + 0.183·47-s + 2.20·49-s + 0.0295·51-s + 1.50·53-s − 0.0486·55-s + 0.346·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.281066291\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.281066291\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 0.568T + 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 0.360T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 0.370T + 17T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 29 | \( 1 - 0.939T + 29T^{2} \) |
| 31 | \( 1 + 9.66T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 - 9.71T + 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.745T + 73T^{2} \) |
| 79 | \( 1 + 0.415T + 79T^{2} \) |
| 83 | \( 1 - 9.26T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 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
−9.109959738712378489852191146317, −8.693304939406647064969092670324, −7.66031533688569428577097904332, −6.86974555955036521775890010158, −6.03835507846753314531492757952, −5.45116032660235916260362165796, −3.78412579682264420598315908340, −3.50498762785343705309704084704, −2.54526778852463296582397838950, −0.73930114888484845428126629533,
0.73930114888484845428126629533, 2.54526778852463296582397838950, 3.50498762785343705309704084704, 3.78412579682264420598315908340, 5.45116032660235916260362165796, 6.03835507846753314531492757952, 6.86974555955036521775890010158, 7.66031533688569428577097904332, 8.693304939406647064969092670324, 9.109959738712378489852191146317