| L(s) = 1 | − 3-s − 1.79·5-s − 4.44·7-s + 9-s − 1.81·11-s − 2.51·13-s + 1.79·15-s + 2.41·17-s − 3.57·19-s + 4.44·21-s − 5.16·23-s − 1.79·25-s − 27-s + 7.45·29-s − 1.28·31-s + 1.81·33-s + 7.95·35-s − 9.97·37-s + 2.51·39-s − 10.2·41-s − 11.0·43-s − 1.79·45-s − 10.6·47-s + 12.7·49-s − 2.41·51-s − 5.98·53-s + 3.24·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.800·5-s − 1.67·7-s + 0.333·9-s − 0.546·11-s − 0.697·13-s + 0.462·15-s + 0.586·17-s − 0.819·19-s + 0.969·21-s − 1.07·23-s − 0.358·25-s − 0.192·27-s + 1.38·29-s − 0.231·31-s + 0.315·33-s + 1.34·35-s − 1.64·37-s + 0.402·39-s − 1.60·41-s − 1.68·43-s − 0.266·45-s − 1.55·47-s + 1.81·49-s − 0.338·51-s − 0.822·53-s + 0.437·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01782626946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01782626946\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 1.79T + 5T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + 2.51T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 + 3.57T + 19T^{2} \) |
| 23 | \( 1 + 5.16T + 23T^{2} \) |
| 29 | \( 1 - 7.45T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 + 9.97T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 5.98T + 53T^{2} \) |
| 59 | \( 1 + 0.0586T + 59T^{2} \) |
| 61 | \( 1 + 4.61T + 61T^{2} \) |
| 67 | \( 1 + 3.96T + 67T^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 - 3.78T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 3.96T + 89T^{2} \) |
| 97 | \( 1 + 7.68T + 97T^{2} \) |
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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
−8.108904249092507928066209693757, −7.18044252528777596336990806612, −6.67801859709994694745932325903, −6.07450522423993754280243657945, −5.20338819350494785960608851004, −4.46730946043369063297354432644, −3.51309299786287690127650814037, −3.06575971859262715254159409670, −1.81864864962515369502170169748, −0.06798201048333432450694230362,
0.06798201048333432450694230362, 1.81864864962515369502170169748, 3.06575971859262715254159409670, 3.51309299786287690127650814037, 4.46730946043369063297354432644, 5.20338819350494785960608851004, 6.07450522423993754280243657945, 6.67801859709994694745932325903, 7.18044252528777596336990806612, 8.108904249092507928066209693757
