L(s) = 1 | − 4-s − 4·5-s + 16-s + 16·19-s + 4·20-s + 11·25-s − 16·29-s + 8·31-s + 24·41-s − 49-s − 16·59-s − 12·61-s − 64-s − 16·76-s − 16·79-s − 4·80-s + 8·89-s − 64·95-s − 11·100-s + 24·101-s − 4·109-s + 16·116-s − 22·121-s − 8·124-s − 24·125-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1.78·5-s + 1/4·16-s + 3.67·19-s + 0.894·20-s + 11/5·25-s − 2.97·29-s + 1.43·31-s + 3.74·41-s − 1/7·49-s − 2.08·59-s − 1.53·61-s − 1/8·64-s − 1.83·76-s − 1.80·79-s − 0.447·80-s + 0.847·89-s − 6.56·95-s − 1.09·100-s + 2.38·101-s − 0.383·109-s + 1.48·116-s − 2·121-s − 0.718·124-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.069783821\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.069783821\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.91946341175581805790662520964, −10.49839725824576295199357472012, −9.750325669363806175511469432261, −9.375143355219890940722532391734, −9.231004043744810732276820062365, −8.710471852531880161530895307943, −7.86094990335840108523802619017, −7.70978173658112462198926068231, −7.40815005074694681976714668290, −7.29784820732422506511104865715, −6.16337119095468751719743091885, −5.86477934218199323320095027219, −5.16243260990912172191610139147, −4.83849137185205450889737786480, −4.09237732048846523750102007496, −3.85889533379040075872733995812, −3.06440711733884952843546864526, −2.91353021490250363503403738067, −1.40245221211779527951862810289, −0.63892303427725341813276550352,
0.63892303427725341813276550352, 1.40245221211779527951862810289, 2.91353021490250363503403738067, 3.06440711733884952843546864526, 3.85889533379040075872733995812, 4.09237732048846523750102007496, 4.83849137185205450889737786480, 5.16243260990912172191610139147, 5.86477934218199323320095027219, 6.16337119095468751719743091885, 7.29784820732422506511104865715, 7.40815005074694681976714668290, 7.70978173658112462198926068231, 7.86094990335840108523802619017, 8.710471852531880161530895307943, 9.231004043744810732276820062365, 9.375143355219890940722532391734, 9.750325669363806175511469432261, 10.49839725824576295199357472012, 10.91946341175581805790662520964