| L(s) = 1 | + 68·3-s + 1.51e3·5-s + 1.02e4·7-s − 1.50e4·9-s + 3.91e3·11-s − 1.76e5·13-s + 1.02e5·15-s + 1.48e5·17-s + 4.99e5·19-s + 6.96e5·21-s − 1.88e6·23-s + 3.26e5·25-s − 2.36e6·27-s − 9.20e5·29-s + 1.37e6·31-s + 2.66e5·33-s + 1.54e7·35-s + 5.06e6·37-s − 1.20e7·39-s − 2.41e7·41-s + 2.57e7·43-s − 2.27e7·45-s − 6.07e7·47-s + 6.46e7·49-s + 1.00e7·51-s + 2.94e7·53-s + 5.91e6·55-s + ⋯ |
| L(s) = 1 | + 0.484·3-s + 1.08·5-s + 1.61·7-s − 0.765·9-s + 0.0806·11-s − 1.71·13-s + 0.523·15-s + 0.430·17-s + 0.879·19-s + 0.781·21-s − 1.40·23-s + 0.167·25-s − 0.855·27-s − 0.241·29-s + 0.268·31-s + 0.0390·33-s + 1.74·35-s + 0.444·37-s − 0.831·39-s − 1.33·41-s + 1.15·43-s − 0.826·45-s − 1.81·47-s + 1.60·49-s + 0.208·51-s + 0.513·53-s + 0.0871·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.979806665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979806665\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 68 T + p^{9} T^{2} \) |
| 5 | \( 1 - 302 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 1464 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 356 p T + p^{9} T^{2} \) |
| 13 | \( 1 + 176594 T + p^{9} T^{2} \) |
| 17 | \( 1 - 148370 T + p^{9} T^{2} \) |
| 19 | \( 1 - 499796 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1889768 T + p^{9} T^{2} \) |
| 29 | \( 1 + 920898 T + p^{9} T^{2} \) |
| 31 | \( 1 - 1379360 T + p^{9} T^{2} \) |
| 37 | \( 1 - 5064966 T + p^{9} T^{2} \) |
| 41 | \( 1 + 24100758 T + p^{9} T^{2} \) |
| 43 | \( 1 - 25785196 T + p^{9} T^{2} \) |
| 47 | \( 1 + 60790224 T + p^{9} T^{2} \) |
| 53 | \( 1 - 29496214 T + p^{9} T^{2} \) |
| 59 | \( 1 - 51819388 T + p^{9} T^{2} \) |
| 61 | \( 1 - 33426910 T + p^{9} T^{2} \) |
| 67 | \( 1 - 144856196 T + p^{9} T^{2} \) |
| 71 | \( 1 - 68397128 T + p^{9} T^{2} \) |
| 73 | \( 1 - 168216202 T + p^{9} T^{2} \) |
| 79 | \( 1 - 235398736 T + p^{9} T^{2} \) |
| 83 | \( 1 + 64639852 T + p^{9} T^{2} \) |
| 89 | \( 1 + 78782694 T + p^{9} T^{2} \) |
| 97 | \( 1 + 24113566 T + p^{9} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.91945850184182812710697230205, −17.97454881562412988200361418686, −17.10343520357264605015720414343, −14.65879360354467059714064454354, −13.97844638654680904383703005425, −11.75939171102679351923867172271, −9.768946281579728479361920887645, −7.958477089867754117418590143370, −5.29155453932789914707898677530, −2.11546748114588182149407175023,
2.11546748114588182149407175023, 5.29155453932789914707898677530, 7.958477089867754117418590143370, 9.768946281579728479361920887645, 11.75939171102679351923867172271, 13.97844638654680904383703005425, 14.65879360354467059714064454354, 17.10343520357264605015720414343, 17.97454881562412988200361418686, 19.91945850184182812710697230205
