| 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
−7.84583825736265941349890868972, −7.32928623626436059506430199840, −6.69890920158193646443230069677, −5.55847498604016471058701597334, −4.89768241245984781937749259028, −4.17747489667715228070708830048, −3.07951569580660602520872948279, −1.99944788838886822138606777297, −1.20922701422812036806202504091, 0,
1.20922701422812036806202504091, 1.99944788838886822138606777297, 3.07951569580660602520872948279, 4.17747489667715228070708830048, 4.89768241245984781937749259028, 5.55847498604016471058701597334, 6.69890920158193646443230069677, 7.32928623626436059506430199840, 7.84583825736265941349890868972
