| L(s) = 1 | − 2.73·2-s − 3-s + 5.50·4-s − 3.38·5-s + 2.73·6-s − 9.61·8-s + 9-s + 9.26·10-s + 1.88·11-s − 5.50·12-s + 1.76·13-s + 3.38·15-s + 15.3·16-s − 5.89·17-s − 2.73·18-s − 6.62·19-s − 18.6·20-s − 5.16·22-s + 23-s + 9.61·24-s + 6.43·25-s − 4.84·26-s − 27-s + 8.24·29-s − 9.26·30-s + 2.94·31-s − 22.7·32-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.577·3-s + 2.75·4-s − 1.51·5-s + 1.11·6-s − 3.39·8-s + 0.333·9-s + 2.93·10-s + 0.567·11-s − 1.58·12-s + 0.490·13-s + 0.873·15-s + 3.82·16-s − 1.42·17-s − 0.645·18-s − 1.52·19-s − 4.16·20-s − 1.10·22-s + 0.208·23-s + 1.96·24-s + 1.28·25-s − 0.950·26-s − 0.192·27-s + 1.53·29-s − 1.69·30-s + 0.528·31-s − 4.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2204512213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2204512213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 11 | \( 1 - 1.88T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 19 | \( 1 + 6.62T + 19T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 2.94T + 31T^{2} \) |
| 37 | \( 1 + 4.43T + 37T^{2} \) |
| 41 | \( 1 + 5.90T + 41T^{2} \) |
| 43 | \( 1 - 0.669T + 43T^{2} \) |
| 47 | \( 1 - 1.79T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 - 1.36T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 + 8.17T + 67T^{2} \) |
| 71 | \( 1 + 0.295T + 71T^{2} \) |
| 73 | \( 1 + 4.45T + 73T^{2} \) |
| 79 | \( 1 + 3.02T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.585963158331735619003293783371, −8.148798702047248372750663148425, −7.21834657040902517983387883389, −6.64932810755966003178867907357, −6.23100932261127160950241437553, −4.68436725343949250436231816429, −3.83291593272185651269079036050, −2.70470802211457488615488536910, −1.52915434460346669921192168325, −0.39250097494924073764365974975,
0.39250097494924073764365974975, 1.52915434460346669921192168325, 2.70470802211457488615488536910, 3.83291593272185651269079036050, 4.68436725343949250436231816429, 6.23100932261127160950241437553, 6.64932810755966003178867907357, 7.21834657040902517983387883389, 8.148798702047248372750663148425, 8.585963158331735619003293783371
