| L(s) = 1 | + 3·9-s + 2·11-s + 10·19-s + 24·29-s + 2·31-s − 40·41-s − 2·49-s + 6·59-s − 6·61-s + 64·71-s − 22·79-s + 9·81-s − 18·89-s + 6·99-s + 26·101-s + 22·109-s + 23·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + ⋯ |
| L(s) = 1 | + 9-s + 0.603·11-s + 2.29·19-s + 4.45·29-s + 0.359·31-s − 6.24·41-s − 2/7·49-s + 0.781·59-s − 0.768·61-s + 7.59·71-s − 2.47·79-s + 81-s − 1.90·89-s + 0.603·99-s + 2.58·101-s + 2.10·109-s + 2.09·121-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 + 3.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.442327597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.442327597\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 93 T^{2} + 6440 T^{4} + 93 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.84444616856926999193497011846, −6.66382406787934641398845607328, −6.49682826297776569943999222570, −6.33197865983497823130620806715, −6.14879860664653064886399727095, −5.57693611473715008357153401594, −5.35007060963551862725872608743, −5.22996115172301160884515969663, −5.01143669925527556259083708861, −4.97111835481233052755104832977, −4.62189654409106141935307866709, −4.46947973162141114562289380629, −4.09117886390434396199635671697, −3.81222719987911031909204887147, −3.47061251594437946512355006275, −3.43610676909688336185443448461, −3.13553626190755665717283993062, −2.81698045225542291486480440355, −2.74268268111958846717805403624, −2.09541406879392420299476405459, −1.73749201477878856878589822295, −1.73299149040166484438055532257, −1.17823540563910402632304150023, −0.75232851537643708635736921052, −0.67903937486665138579953681563,
0.67903937486665138579953681563, 0.75232851537643708635736921052, 1.17823540563910402632304150023, 1.73299149040166484438055532257, 1.73749201477878856878589822295, 2.09541406879392420299476405459, 2.74268268111958846717805403624, 2.81698045225542291486480440355, 3.13553626190755665717283993062, 3.43610676909688336185443448461, 3.47061251594437946512355006275, 3.81222719987911031909204887147, 4.09117886390434396199635671697, 4.46947973162141114562289380629, 4.62189654409106141935307866709, 4.97111835481233052755104832977, 5.01143669925527556259083708861, 5.22996115172301160884515969663, 5.35007060963551862725872608743, 5.57693611473715008357153401594, 6.14879860664653064886399727095, 6.33197865983497823130620806715, 6.49682826297776569943999222570, 6.66382406787934641398845607328, 6.84444616856926999193497011846
