| L(s) = 1 | − 4-s + 9-s − 11-s + 16-s − 7·23-s + 2·31-s − 36-s + 3·37-s + 44-s − 47-s − 6·49-s + 13·53-s − 6·59-s − 64-s + 19·67-s + 15·71-s + 81-s − 9·89-s + 7·92-s − 5·97-s − 99-s − 14·103-s − 21·113-s − 10·121-s − 2·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/3·9-s − 0.301·11-s + 1/4·16-s − 1.45·23-s + 0.359·31-s − 1/6·36-s + 0.493·37-s + 0.150·44-s − 0.145·47-s − 6/7·49-s + 1.78·53-s − 0.781·59-s − 1/8·64-s + 2.32·67-s + 1.78·71-s + 1/9·81-s − 0.953·89-s + 0.729·92-s − 0.507·97-s − 0.100·99-s − 1.37·103-s − 1.97·113-s − 0.909·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
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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
−7.49218692038248829355441858591, −6.91284135376468455973983410400, −6.56471585686373781253316256995, −6.20182513688504551861908390508, −5.58692850680909792869153854388, −5.30585349915632977018317454014, −4.85593399741646832485196909844, −4.30310042855857364820712061170, −3.83463620163683681925086085033, −3.65539067613170470704993115201, −2.73146351234268196468991586384, −2.40235627212164701945417116949, −1.67625184656289216525805306861, −0.942511437481046915555847362770, 0,
0.942511437481046915555847362770, 1.67625184656289216525805306861, 2.40235627212164701945417116949, 2.73146351234268196468991586384, 3.65539067613170470704993115201, 3.83463620163683681925086085033, 4.30310042855857364820712061170, 4.85593399741646832485196909844, 5.30585349915632977018317454014, 5.58692850680909792869153854388, 6.20182513688504551861908390508, 6.56471585686373781253316256995, 6.91284135376468455973983410400, 7.49218692038248829355441858591
