| L(s) = 1 | + 16·2-s − 273·3-s + 256·4-s − 1.01e3·5-s − 4.36e3·6-s − 3.95e3·7-s + 4.09e3·8-s + 5.48e4·9-s − 1.62e4·10-s + 5.09e4·11-s − 6.98e4·12-s − 6.32e4·14-s + 2.77e5·15-s + 6.55e4·16-s + 5.09e5·17-s + 8.77e5·18-s + 6.26e5·19-s − 2.59e5·20-s + 1.07e6·21-s + 8.15e5·22-s + 6.53e5·23-s − 1.11e6·24-s − 9.22e5·25-s − 9.59e6·27-s − 1.01e6·28-s − 4.94e6·29-s + 4.43e6·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.94·3-s + 1/2·4-s − 0.726·5-s − 1.37·6-s − 0.622·7-s + 0.353·8-s + 2.78·9-s − 0.513·10-s + 1.05·11-s − 0.972·12-s − 0.440·14-s + 1.41·15-s + 1/4·16-s + 1.48·17-s + 1.97·18-s + 1.10·19-s − 0.363·20-s + 1.21·21-s + 0.742·22-s + 0.486·23-s − 0.687·24-s − 0.472·25-s − 3.47·27-s − 0.311·28-s − 1.29·29-s + 0.999·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.509486585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.509486585\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 91 p T + p^{9} T^{2} \) |
| 5 | \( 1 + 203 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 565 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 50998 T + p^{9} T^{2} \) |
| 17 | \( 1 - 509757 T + p^{9} T^{2} \) |
| 19 | \( 1 - 626574 T + p^{9} T^{2} \) |
| 23 | \( 1 - 653524 T + p^{9} T^{2} \) |
| 29 | \( 1 + 4943006 T + p^{9} T^{2} \) |
| 31 | \( 1 + 4071700 T + p^{9} T^{2} \) |
| 37 | \( 1 + 2348883 T + p^{9} T^{2} \) |
| 41 | \( 1 - 13350960 T + p^{9} T^{2} \) |
| 43 | \( 1 + 7834847 T + p^{9} T^{2} \) |
| 47 | \( 1 - 39637681 T + p^{9} T^{2} \) |
| 53 | \( 1 - 73200924 T + p^{9} T^{2} \) |
| 59 | \( 1 - 141141614 T + p^{9} T^{2} \) |
| 61 | \( 1 + 132061256 T + p^{9} T^{2} \) |
| 67 | \( 1 - 185673110 T + p^{9} T^{2} \) |
| 71 | \( 1 + 224452625 T + p^{9} T^{2} \) |
| 73 | \( 1 - 2363338 p T + p^{9} T^{2} \) |
| 79 | \( 1 + 643288156 T + p^{9} T^{2} \) |
| 83 | \( 1 + 720077280 T + p^{9} T^{2} \) |
| 89 | \( 1 - 73028106 T + p^{9} T^{2} \) |
| 97 | \( 1 - 15879778 T + p^{9} T^{2} \) |
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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
−10.26816822972103868555994121455, −9.455000979266892719060184805722, −7.48716594715266118519925592368, −6.99929639864244902410519257413, −5.85847312183017710475032189868, −5.41310590195499113512283772360, −4.16400916500400204301550399584, −3.50271629809486983932378229789, −1.44114462605644691857643339500, −0.57674611145920558468241113567,
0.57674611145920558468241113567, 1.44114462605644691857643339500, 3.50271629809486983932378229789, 4.16400916500400204301550399584, 5.41310590195499113512283772360, 5.85847312183017710475032189868, 6.99929639864244902410519257413, 7.48716594715266118519925592368, 9.455000979266892719060184805722, 10.26816822972103868555994121455
