| L(s) = 1 | − 60·11-s + 56·19-s + 420·29-s − 8·31-s − 480·41-s + 682·49-s − 900·59-s − 332·61-s + 2.04e3·71-s + 1.83e3·79-s − 840·89-s − 900·101-s + 3.40e3·109-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.37e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.64·11-s + 0.676·19-s + 2.68·29-s − 0.0463·31-s − 1.82·41-s + 1.98·49-s − 1.98·59-s − 0.696·61-s + 3.40·71-s + 2.60·79-s − 1.00·89-s − 0.886·101-s + 2.99·109-s + 0.0285·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.99·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.397536694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.397536694\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4378 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1726 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 210 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 61306 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 140518 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 193246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 296854 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 450 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 166 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 222938 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1020 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 715534 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 916 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 156026 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 420 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 540098 T^{2} + p^{6} T^{4} \) |
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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
−9.859958741930050112163421641324, −9.669766547832549444602307137189, −9.108312963500059114962368417645, −8.560655716411356922320459271382, −8.118443000210158932411939371525, −8.045841514149026019984096396970, −7.39385929967941870811973318823, −6.99478329200163005348377374548, −6.48105692284846780380498347276, −6.12306024057428451509262079667, −5.36829310564883795972252311327, −5.17244807287499468475779863356, −4.72278427671724149084418650943, −4.20270811482181738308586973216, −3.35207164519829761499528186795, −3.08247278493990379848327026725, −2.45856659559515618628989173959, −1.94750423049145117134403054181, −0.984316828859506754651978103772, −0.47272824774670665840605765436,
0.47272824774670665840605765436, 0.984316828859506754651978103772, 1.94750423049145117134403054181, 2.45856659559515618628989173959, 3.08247278493990379848327026725, 3.35207164519829761499528186795, 4.20270811482181738308586973216, 4.72278427671724149084418650943, 5.17244807287499468475779863356, 5.36829310564883795972252311327, 6.12306024057428451509262079667, 6.48105692284846780380498347276, 6.99478329200163005348377374548, 7.39385929967941870811973318823, 8.045841514149026019984096396970, 8.118443000210158932411939371525, 8.560655716411356922320459271382, 9.108312963500059114962368417645, 9.669766547832549444602307137189, 9.859958741930050112163421641324
