| L(s) = 1 | − 4·5-s + 4·11-s + 6·19-s + 11·25-s + 12·29-s + 2·31-s − 20·41-s + 10·49-s − 16·55-s − 20·59-s + 10·61-s + 12·71-s − 30·79-s − 4·89-s − 24·95-s + 32·101-s + 22·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + 149-s + 151-s − 8·155-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.20·11-s + 1.37·19-s + 11/5·25-s + 2.22·29-s + 0.359·31-s − 3.12·41-s + 10/7·49-s − 2.15·55-s − 2.60·59-s + 1.28·61-s + 1.42·71-s − 3.37·79-s − 0.423·89-s − 2.46·95-s + 3.18·101-s + 2.10·109-s − 0.909·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + 0.0819·149-s + 0.0813·151-s − 0.642·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.311997855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311997855\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.38515460673658854886615464507, −10.54939732165623187725831600614, −10.12912540563452955944081984675, −9.907555653910551580120790769048, −9.079301038216793549547981869143, −8.685923469006776668757095878589, −8.485715944527938870052490067798, −7.932628517368710060976294001566, −7.35246687627331741518944103509, −7.14212536377970532250190407346, −6.53018612404254597605459097087, −6.21835325972148844419345086621, −5.24997908052656329581298993067, −4.86582177645926965376547087074, −4.33121207109617791024599879894, −3.82256187758409686623163705284, −3.23736979727494902191180070795, −2.89484195594571694937824998390, −1.55404405174510004775783383457, −0.73024365488627143114837307951,
0.73024365488627143114837307951, 1.55404405174510004775783383457, 2.89484195594571694937824998390, 3.23736979727494902191180070795, 3.82256187758409686623163705284, 4.33121207109617791024599879894, 4.86582177645926965376547087074, 5.24997908052656329581298993067, 6.21835325972148844419345086621, 6.53018612404254597605459097087, 7.14212536377970532250190407346, 7.35246687627331741518944103509, 7.932628517368710060976294001566, 8.485715944527938870052490067798, 8.685923469006776668757095878589, 9.079301038216793549547981869143, 9.907555653910551580120790769048, 10.12912540563452955944081984675, 10.54939732165623187725831600614, 11.38515460673658854886615464507
