| L(s) = 1 | + 38·11-s + 182·19-s − 544·29-s − 460·31-s − 234·41-s + 650·49-s + 624·59-s + 340·61-s + 104·71-s − 2.10e3·79-s + 1.59e3·89-s − 972·101-s − 252·109-s − 1.57e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.25e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.04·11-s + 2.19·19-s − 3.48·29-s − 2.66·31-s − 0.891·41-s + 1.89·49-s + 1.37·59-s + 0.713·61-s + 0.173·71-s − 3.00·79-s + 1.90·89-s − 0.957·101-s − 0.221·109-s − 1.18·121-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 + 1.93·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.304171411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304171411\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 650 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 19 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4201 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5942 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 272 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 230 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 68182 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 117 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 20630 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 204942 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 136150 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 312 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 170 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 19357 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 52 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 184327 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1054 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1020373 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 799 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 899902 T^{2} + p^{6} T^{4} \) |
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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
−9.241685726792661589295040667046, −8.894693498234480426518741133861, −8.406585703449442169131287206968, −7.76627119277445115009282800726, −7.41148629895194224364889449462, −7.13608954064081721106802313744, −7.05601620856204670762613801380, −6.26813029503169776218473653404, −5.68666198437608575026678268228, −5.42103961117880266496177530750, −5.39500464243317504707125112597, −4.59445979942090984640690201319, −3.83886667775980941876991893183, −3.62969378805843854786234286014, −3.56289228253312399108793702220, −2.62386882359681457512083123882, −2.07958752919031604940185233843, −1.50806694959339091488919907769, −1.15240306522324063746122816158, −0.24916026252413144721109458942,
0.24916026252413144721109458942, 1.15240306522324063746122816158, 1.50806694959339091488919907769, 2.07958752919031604940185233843, 2.62386882359681457512083123882, 3.56289228253312399108793702220, 3.62969378805843854786234286014, 3.83886667775980941876991893183, 4.59445979942090984640690201319, 5.39500464243317504707125112597, 5.42103961117880266496177530750, 5.68666198437608575026678268228, 6.26813029503169776218473653404, 7.05601620856204670762613801380, 7.13608954064081721106802313744, 7.41148629895194224364889449462, 7.76627119277445115009282800726, 8.406585703449442169131287206968, 8.894693498234480426518741133861, 9.241685726792661589295040667046
