| L(s) = 1 | − 6.56e3·3-s − 1.60e6·5-s − 9.41e6·7-s + 4.30e7·9-s − 1.86e8·11-s − 2.62e9·13-s + 1.05e10·15-s + 4.37e10·17-s − 9.65e10·19-s + 6.17e10·21-s + 2.90e11·23-s + 1.82e12·25-s − 2.82e11·27-s + 1.39e12·29-s + 7.64e12·31-s + 1.22e12·33-s + 1.51e13·35-s − 3.33e13·37-s + 1.72e13·39-s − 1.20e13·41-s − 7.55e11·43-s − 6.92e13·45-s − 2.80e14·47-s − 1.43e14·49-s − 2.87e14·51-s + 4.60e14·53-s + 3.00e14·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.84·5-s − 0.617·7-s + 1/3·9-s − 0.262·11-s − 0.892·13-s + 1.06·15-s + 1.52·17-s − 1.30·19-s + 0.356·21-s + 0.774·23-s + 2.39·25-s − 0.192·27-s + 0.519·29-s + 1.61·31-s + 0.151·33-s + 1.13·35-s − 1.56·37-s + 0.515·39-s − 0.235·41-s − 0.00985·43-s − 0.614·45-s − 1.71·47-s − 0.618·49-s − 0.878·51-s + 1.01·53-s + 0.484·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.6147040071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6147040071\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{8} T \) |
| good | 5 | \( 1 + 321786 p T + p^{17} T^{2} \) |
| 7 | \( 1 + 1345312 p T + p^{17} T^{2} \) |
| 11 | \( 1 + 186910524 T + p^{17} T^{2} \) |
| 13 | \( 1 + 201957130 p T + p^{17} T^{2} \) |
| 17 | \( 1 - 43782311106 T + p^{17} T^{2} \) |
| 19 | \( 1 + 96594985540 T + p^{17} T^{2} \) |
| 23 | \( 1 - 290867937336 T + p^{17} T^{2} \) |
| 29 | \( 1 - 1398617429094 T + p^{17} T^{2} \) |
| 31 | \( 1 - 7647898359464 T + p^{17} T^{2} \) |
| 37 | \( 1 + 33369516616762 T + p^{17} T^{2} \) |
| 41 | \( 1 + 12032733393990 T + p^{17} T^{2} \) |
| 43 | \( 1 + 755092495804 T + p^{17} T^{2} \) |
| 47 | \( 1 + 280540358127936 T + p^{17} T^{2} \) |
| 53 | \( 1 - 460570203615582 T + p^{17} T^{2} \) |
| 59 | \( 1 - 1078467799153284 T + p^{17} T^{2} \) |
| 61 | \( 1 + 1980778975313218 T + p^{17} T^{2} \) |
| 67 | \( 1 - 4850190377589884 T + p^{17} T^{2} \) |
| 71 | \( 1 - 2707574704052040 T + p^{17} T^{2} \) |
| 73 | \( 1 + 5002264428090742 T + p^{17} T^{2} \) |
| 79 | \( 1 + 9774477292907752 T + p^{17} T^{2} \) |
| 83 | \( 1 - 17112919183614396 T + p^{17} T^{2} \) |
| 89 | \( 1 - 34698182155846650 T + p^{17} T^{2} \) |
| 97 | \( 1 - 68616916871806082 T + p^{17} T^{2} \) |
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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
−15.94683915833193204961245522791, −14.87982724463990142315989459613, −12.61065100524886002050016915239, −11.78145360328474095800241457065, −10.29624854251689873816032104134, −8.178098904369178098629047071150, −6.87143389004008390926565509701, −4.77388497118489147225673141412, −3.28754307866037394997164681064, −0.52787478244290505466548330523,
0.52787478244290505466548330523, 3.28754307866037394997164681064, 4.77388497118489147225673141412, 6.87143389004008390926565509701, 8.178098904369178098629047071150, 10.29624854251689873816032104134, 11.78145360328474095800241457065, 12.61065100524886002050016915239, 14.87982724463990142315989459613, 15.94683915833193204961245522791
