| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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
−16.74694518798259763117751377310, −14.89771427542189998042614875433, −13.38456992068189265273424758871, −12.26096501403816484833617051809, −10.32970550536575136246635104335, −9.280050381485872692965523792209, −6.75084341655681849975423585181, −5.46200897361597409347253644663, −2.72183064437687440973998944660, 0,
2.72183064437687440973998944660, 5.46200897361597409347253644663, 6.75084341655681849975423585181, 9.280050381485872692965523792209, 10.32970550536575136246635104335, 12.26096501403816484833617051809, 13.38456992068189265273424758871, 14.89771427542189998042614875433, 16.74694518798259763117751377310
