| L(s) = 1 | − 2·5-s − 9-s + 6·11-s − 8·19-s − 25-s − 16·29-s − 8·31-s − 10·41-s + 2·45-s + 13·49-s − 12·55-s − 24·59-s − 30·61-s + 30·71-s − 10·79-s + 81-s − 10·89-s + 16·95-s − 6·99-s + 20·101-s − 8·109-s + 5·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/3·9-s + 1.80·11-s − 1.83·19-s − 1/5·25-s − 2.97·29-s − 1.43·31-s − 1.56·41-s + 0.298·45-s + 13/7·49-s − 1.61·55-s − 3.12·59-s − 3.84·61-s + 3.56·71-s − 1.12·79-s + 1/9·81-s − 1.05·89-s + 1.64·95-s − 0.603·99-s + 1.99·101-s − 0.766·109-s + 5/11·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−8.584567358182662148317822308148, −8.207893861561208731284827181138, −7.56954127951197732635338175233, −7.42710962902836852653270993867, −7.20668637132121573682551632286, −6.52823676305371814018197587750, −6.18000428276478447941922373936, −6.07631461952564990074646328580, −5.46514355964137502655914572601, −5.01758163415353955326031926055, −4.43089147607242561421229635600, −4.14884612689699300760991820869, −3.65174825085545839842790305822, −3.61599543221952994061732458542, −2.98626232723233596923955245353, −2.08151118744873257016832198578, −1.84458198943444922102874482455, −1.31470857058652956061762040967, 0, 0,
1.31470857058652956061762040967, 1.84458198943444922102874482455, 2.08151118744873257016832198578, 2.98626232723233596923955245353, 3.61599543221952994061732458542, 3.65174825085545839842790305822, 4.14884612689699300760991820869, 4.43089147607242561421229635600, 5.01758163415353955326031926055, 5.46514355964137502655914572601, 6.07631461952564990074646328580, 6.18000428276478447941922373936, 6.52823676305371814018197587750, 7.20668637132121573682551632286, 7.42710962902836852653270993867, 7.56954127951197732635338175233, 8.207893861561208731284827181138, 8.584567358182662148317822308148
