| L(s) = 1 | − 2·3-s + 7·4-s − 51·9-s − 14·12-s + 52·13-s − 15·16-s − 90·17-s + 324·23-s + 169·25-s + 158·27-s − 288·29-s − 357·36-s − 104·39-s − 194·43-s + 30·48-s + 461·49-s + 180·51-s + 364·52-s − 828·53-s + 752·61-s − 553·64-s − 630·68-s − 648·69-s − 338·75-s − 1.66e3·79-s + 1.86e3·81-s + 576·87-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 7/8·4-s − 1.88·9-s − 0.336·12-s + 1.10·13-s − 0.234·16-s − 1.28·17-s + 2.93·23-s + 1.35·25-s + 1.12·27-s − 1.84·29-s − 1.65·36-s − 0.427·39-s − 0.688·43-s + 0.0902·48-s + 1.34·49-s + 0.494·51-s + 0.970·52-s − 2.14·53-s + 1.57·61-s − 1.08·64-s − 1.12·68-s − 1.13·69-s − 0.520·75-s − 2.36·79-s + 2.56·81-s + 0.709·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8760006070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8760006070\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 169 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 461 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 358 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 45 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13682 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 144 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10114 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 9497 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 100978 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 97 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 195325 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 414 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 138274 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 376 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 600230 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 588373 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 592 T + p^{3} T^{2} )( 1 + 592 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 830 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 951730 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1218094 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1099442 T^{2} + p^{6} T^{4} \) |
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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
−20.07747087618661690760159125164, −19.12066855542506934627592696216, −18.56570016008740859489133774140, −17.62314566303340338455370062924, −17.00472259346844510464561559806, −16.63463237935173972028166248689, −15.74602187090146680512445425353, −15.05289148393711810595773703217, −14.46122281821318661474247024287, −13.40313563714491926733499154473, −12.82368120640982122209391662241, −11.51890229060460885251302265496, −11.04973796206346326445617763542, −10.96032656074757330025078934392, −8.983836264546810122151905930528, −8.704185250031597270937396060796, −7.09714183483057059409876765045, −6.30719010199337973804419825585, −5.18901415598640781945830316564, −2.95310855429547515318810034846,
2.95310855429547515318810034846, 5.18901415598640781945830316564, 6.30719010199337973804419825585, 7.09714183483057059409876765045, 8.704185250031597270937396060796, 8.983836264546810122151905930528, 10.96032656074757330025078934392, 11.04973796206346326445617763542, 11.51890229060460885251302265496, 12.82368120640982122209391662241, 13.40313563714491926733499154473, 14.46122281821318661474247024287, 15.05289148393711810595773703217, 15.74602187090146680512445425353, 16.63463237935173972028166248689, 17.00472259346844510464561559806, 17.62314566303340338455370062924, 18.56570016008740859489133774140, 19.12066855542506934627592696216, 20.07747087618661690760159125164
