| L(s) = 1 | + 1.63·2-s + 0.661·4-s + 1.44·7-s − 2.18·8-s − 0.797·11-s − 3.02·13-s + 2.35·14-s − 4.88·16-s + 0.282·17-s + 2.88·19-s − 1.30·22-s − 3.47·23-s − 4.93·26-s + 0.956·28-s + 7.73·29-s + 2.38·31-s − 3.60·32-s + 0.460·34-s − 4.64·37-s + 4.70·38-s − 8.42·41-s − 6.07·43-s − 0.527·44-s − 5.66·46-s − 1.16·47-s − 4.91·49-s − 2.00·52-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.330·4-s + 0.546·7-s − 0.771·8-s − 0.240·11-s − 0.839·13-s + 0.630·14-s − 1.22·16-s + 0.0684·17-s + 0.661·19-s − 0.277·22-s − 0.724·23-s − 0.968·26-s + 0.180·28-s + 1.43·29-s + 0.428·31-s − 0.637·32-s + 0.0789·34-s − 0.763·37-s + 0.763·38-s − 1.31·41-s − 0.927·43-s − 0.0795·44-s − 0.835·46-s − 0.169·47-s − 0.701·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 + 0.797T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 17 | \( 1 - 0.282T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 + 4.64T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 + 1.16T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 0.868T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 4.63T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.69320241730769910807659927478, −6.91314925746244753417644838479, −6.18268769234620243152864093388, −5.38140263911911144956878262972, −4.84472076784578122093404691993, −4.30059880651034139139831990733, −3.28749156010061210710105282694, −2.68737859777769722134017657121, −1.57200137224191648637412220536, 0,
1.57200137224191648637412220536, 2.68737859777769722134017657121, 3.28749156010061210710105282694, 4.30059880651034139139831990733, 4.84472076784578122093404691993, 5.38140263911911144956878262972, 6.18268769234620243152864093388, 6.91314925746244753417644838479, 7.69320241730769910807659927478
