| L(s) = 1 | − 4·4-s + 16·16-s + 80·29-s + 160·41-s − 98·49-s − 44·61-s − 64·64-s + 320·89-s − 80·101-s − 364·109-s − 320·116-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 640·164-s + 167-s + 238·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s + 2.75·29-s + 3.90·41-s − 2·49-s − 0.721·61-s − 64-s + 3.59·89-s − 0.792·101-s − 3.33·109-s − 2.75·116-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s − 3.90·164-s + 0.00598·167-s + 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.116847265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116847265\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )( 1 + 96 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 160 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} ) \) |
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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
−10.07982587195357525504705340380, −9.586475660256515602254086037809, −9.182783159808963125587651340582, −9.136376250783893696167736405410, −8.258886436307642805109100082212, −8.157639555521874290071132518076, −7.78794276866251576123792979772, −7.19176073931653637401584545613, −6.64465359542742544264425326525, −6.02517159416098460207701642273, −6.01134698761195672394974302811, −5.02596265131659000602478233837, −4.93106542204228106335146193714, −4.15863202857464013306969483543, −4.12815353150842717986766337941, −3.01718158822779872840855952951, −2.95007254879951885496594675353, −2.00085995923988687091411433567, −1.05301674033675730715462372274, −0.59508477684207740752824866212,
0.59508477684207740752824866212, 1.05301674033675730715462372274, 2.00085995923988687091411433567, 2.95007254879951885496594675353, 3.01718158822779872840855952951, 4.12815353150842717986766337941, 4.15863202857464013306969483543, 4.93106542204228106335146193714, 5.02596265131659000602478233837, 6.01134698761195672394974302811, 6.02517159416098460207701642273, 6.64465359542742544264425326525, 7.19176073931653637401584545613, 7.78794276866251576123792979772, 8.157639555521874290071132518076, 8.258886436307642805109100082212, 9.136376250783893696167736405410, 9.182783159808963125587651340582, 9.586475660256515602254086037809, 10.07982587195357525504705340380
