| L(s) = 1 | + 2-s + 4-s − 2.89·5-s + 0.647·7-s + 8-s − 2.89·10-s + 3.72·11-s + 0.715·13-s + 0.647·14-s + 16-s + 0.587·17-s + 6.18·19-s − 2.89·20-s + 3.72·22-s + 7.30·23-s + 3.37·25-s + 0.715·26-s + 0.647·28-s + 0.536·29-s − 8.61·31-s + 32-s + 0.587·34-s − 1.87·35-s − 2.58·37-s + 6.18·38-s − 2.89·40-s − 6.06·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.29·5-s + 0.244·7-s + 0.353·8-s − 0.915·10-s + 1.12·11-s + 0.198·13-s + 0.172·14-s + 0.250·16-s + 0.142·17-s + 1.41·19-s − 0.647·20-s + 0.794·22-s + 1.52·23-s + 0.675·25-s + 0.140·26-s + 0.122·28-s + 0.0995·29-s − 1.54·31-s + 0.176·32-s + 0.100·34-s − 0.316·35-s − 0.424·37-s + 1.00·38-s − 0.457·40-s − 0.946·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.029116593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.029116593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 2.89T + 5T^{2} \) |
| 7 | \( 1 - 0.647T + 7T^{2} \) |
| 11 | \( 1 - 3.72T + 11T^{2} \) |
| 13 | \( 1 - 0.715T + 13T^{2} \) |
| 17 | \( 1 - 0.587T + 17T^{2} \) |
| 19 | \( 1 - 6.18T + 19T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 - 0.536T + 29T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 + 6.06T + 41T^{2} \) |
| 43 | \( 1 - 0.0254T + 43T^{2} \) |
| 47 | \( 1 - 3.03T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 - 3.34T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.55029537311618394711532435867, −7.20114546019648064005071458012, −6.56455601924176335141171426218, −5.55307480191434469547289728929, −4.98950585076641106429965524597, −4.19020224700419524486846000451, −3.52875393812953776133624109794, −3.12042643572685785476342670574, −1.74302256335895659800330226066, −0.810302465441579329971638412352,
0.810302465441579329971638412352, 1.74302256335895659800330226066, 3.12042643572685785476342670574, 3.52875393812953776133624109794, 4.19020224700419524486846000451, 4.98950585076641106429965524597, 5.55307480191434469547289728929, 6.56455601924176335141171426218, 7.20114546019648064005071458012, 7.55029537311618394711532435867
