| L(s) = 1 | + 3-s − 9-s − 4·11-s + 4·13-s − 17-s − 3·19-s − 7·23-s + 11·29-s + 10·31-s − 4·33-s + 3·37-s + 4·39-s + 9·41-s − 9·43-s − 51-s − 6·53-s − 3·57-s + 2·59-s − 14·61-s − 24·67-s − 7·69-s − 9·71-s + 17·73-s + 79-s − 4·81-s − 15·83-s + 11·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.242·17-s − 0.688·19-s − 1.45·23-s + 2.04·29-s + 1.79·31-s − 0.696·33-s + 0.493·37-s + 0.640·39-s + 1.40·41-s − 1.37·43-s − 0.140·51-s − 0.824·53-s − 0.397·57-s + 0.260·59-s − 1.79·61-s − 2.93·67-s − 0.842·69-s − 1.06·71-s + 1.98·73-s + 0.112·79-s − 4/9·81-s − 1.64·83-s + 1.17·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.502620794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502620794\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T - 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 98 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 102 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 24 T + 261 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 17 T + 214 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 120 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15 T + 184 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.86106034830499363535280905924, −7.77754541801266973320485513574, −7.16047932246406090232407885502, −6.81028084083697954922084973542, −6.32980250793905034625729403157, −6.21462683182755426618202483386, −5.83259435824048837652576465824, −5.66456427048430994745195336638, −4.94703115437678994228897046944, −4.51353252517580402730714679205, −4.48669435388522843264942160301, −4.18350765372730375175673674396, −3.32600136283598619836183926162, −3.25190110829429909570949590966, −2.72155501542962927823563927677, −2.63425221541114374765796023429, −1.93592214692765153906388088179, −1.61921784933775463351154997502, −0.919684243844775661589864311299, −0.38207870899316422186901465476,
0.38207870899316422186901465476, 0.919684243844775661589864311299, 1.61921784933775463351154997502, 1.93592214692765153906388088179, 2.63425221541114374765796023429, 2.72155501542962927823563927677, 3.25190110829429909570949590966, 3.32600136283598619836183926162, 4.18350765372730375175673674396, 4.48669435388522843264942160301, 4.51353252517580402730714679205, 4.94703115437678994228897046944, 5.66456427048430994745195336638, 5.83259435824048837652576465824, 6.21462683182755426618202483386, 6.32980250793905034625729403157, 6.81028084083697954922084973542, 7.16047932246406090232407885502, 7.77754541801266973320485513574, 7.86106034830499363535280905924
