| L(s) = 1 | − 12·17-s + 8·19-s + 2·25-s − 12·41-s + 8·43-s − 2·49-s + 24·59-s − 8·67-s − 4·73-s + 12·89-s − 4·97-s + 24·107-s + 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2.91·17-s + 1.83·19-s + 2/5·25-s − 1.87·41-s + 1.21·43-s − 2/7·49-s + 3.12·59-s − 0.977·67-s − 0.468·73-s + 1.27·89-s − 0.406·97-s + 2.32·107-s + 1.12·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.718560367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718560367\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.051958995200110935297158929877, −8.872238150144058509099300033791, −8.393135543397082230969700621486, −8.297128464064606045304138325176, −7.40784544888046377444052101226, −7.33661936490695083674831742076, −6.87296624703370658242518001585, −6.63533842994695874337372444054, −6.00748604342774431415303044952, −5.78738706852668711508938914379, −5.05640537320441894785617631505, −4.89148071841182910687091315400, −4.47073029822226100270455099896, −3.88816374686518382818613341114, −3.50402801172082537256356145911, −2.95041560502240512540940991021, −2.34522358440884180999113886745, −2.04697187072846738510564596377, −1.24474527390506146238494436048, −0.47076910174565786246861024592,
0.47076910174565786246861024592, 1.24474527390506146238494436048, 2.04697187072846738510564596377, 2.34522358440884180999113886745, 2.95041560502240512540940991021, 3.50402801172082537256356145911, 3.88816374686518382818613341114, 4.47073029822226100270455099896, 4.89148071841182910687091315400, 5.05640537320441894785617631505, 5.78738706852668711508938914379, 6.00748604342774431415303044952, 6.63533842994695874337372444054, 6.87296624703370658242518001585, 7.33661936490695083674831742076, 7.40784544888046377444052101226, 8.297128464064606045304138325176, 8.393135543397082230969700621486, 8.872238150144058509099300033791, 9.051958995200110935297158929877
