| L(s) = 1 | − 2-s − 6·3-s − 7·4-s + 6·6-s − 2·7-s + 7·8-s + 27·9-s + 22·11-s + 42·12-s + 76·13-s + 2·14-s − 7·16-s + 26·17-s − 27·18-s − 54·19-s + 12·21-s − 22·22-s − 224·23-s − 42·24-s − 76·26-s − 108·27-s + 14·28-s + 222·29-s − 40·31-s + 71·32-s − 132·33-s − 26·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.15·3-s − 7/8·4-s + 0.408·6-s − 0.107·7-s + 0.309·8-s + 9-s + 0.603·11-s + 1.01·12-s + 1.62·13-s + 0.0381·14-s − 0.109·16-s + 0.370·17-s − 0.353·18-s − 0.652·19-s + 0.124·21-s − 0.213·22-s − 2.03·23-s − 0.357·24-s − 0.573·26-s − 0.769·27-s + 0.0944·28-s + 1.42·29-s − 0.231·31-s + 0.392·32-s − 0.696·33-s − 0.131·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p^{3} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 654 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 76 T + 5310 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 26 T + 2570 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 54 T + 11774 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 222 T + 43642 T^{2} - 222 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 40 T - 29250 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 48 T + 85910 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 494 T + 198818 T^{2} + 494 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 66 T + 99086 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 64 T + 189662 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 84 T + 164350 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 196 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 1104 T + 736358 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 928 T + 626214 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 456 T + 488494 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 592 T + 341742 T^{2} - 592 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 230 T + 954126 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 348 T + 307798 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 972 T + 1645606 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1184 T + 720510 T^{2} - 1184 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.516634969959131712289912823603, −9.388783713657521336350301546353, −8.617047255971672098442646368821, −8.538905638022073538242109058541, −8.017559095833157817522963426089, −7.60468890540970895348702857243, −6.81991248830890577622727877409, −6.35526429927175196195866002037, −6.24839944376472449862156930128, −5.83669408856165585027172253858, −5.01013451756560889240117851100, −4.82552399191689847441390459767, −4.12703448592094623844671868544, −3.84415962003726466753577053087, −3.30758697277232841649373431549, −2.30085612818934063010651646502, −1.43400480599740655191800978289, −1.14259457012584259264969040226, 0, 0,
1.14259457012584259264969040226, 1.43400480599740655191800978289, 2.30085612818934063010651646502, 3.30758697277232841649373431549, 3.84415962003726466753577053087, 4.12703448592094623844671868544, 4.82552399191689847441390459767, 5.01013451756560889240117851100, 5.83669408856165585027172253858, 6.24839944376472449862156930128, 6.35526429927175196195866002037, 6.81991248830890577622727877409, 7.60468890540970895348702857243, 8.017559095833157817522963426089, 8.538905638022073538242109058541, 8.617047255971672098442646368821, 9.388783713657521336350301546353, 9.516634969959131712289912823603
