| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 3.41·11-s + 3.26·13-s + 15-s − 6.68·17-s + 7.41·19-s + 21-s − 2.15·23-s + 25-s + 27-s − 6.68·29-s + 9.57·31-s − 3.41·33-s + 35-s + 6.68·37-s + 3.26·39-s + 8.83·41-s + 8.68·43-s + 45-s + 2.15·47-s + 49-s − 6.68·51-s + 1.41·53-s − 3.41·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s − 1.03·11-s + 0.904·13-s + 0.258·15-s − 1.62·17-s + 1.70·19-s + 0.218·21-s − 0.449·23-s + 0.200·25-s + 0.192·27-s − 1.24·29-s + 1.71·31-s − 0.595·33-s + 0.169·35-s + 1.09·37-s + 0.522·39-s + 1.38·41-s + 1.32·43-s + 0.149·45-s + 0.314·47-s + 0.142·49-s − 0.935·51-s + 0.194·53-s − 0.460·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.680850419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.680850419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 7.41T + 19T^{2} \) |
| 23 | \( 1 + 2.15T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 9.57T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 - 8.83T + 41T^{2} \) |
| 43 | \( 1 - 8.68T + 43T^{2} \) |
| 47 | \( 1 - 2.15T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 0.156T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 - 0.738T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 4.52T + 89T^{2} \) |
| 97 | \( 1 - 3.26T + 97T^{2} \) |
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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
−8.661618020475539695421864554221, −7.82868470191142154949206770873, −7.38044722573046949148229558965, −6.28288998913628908072305703570, −5.65649319138181882137357518874, −4.72049182926131707319544056283, −3.96604206675158700563114268969, −2.83456106733617816587890482767, −2.20986485036028532578747169119, −0.981093777738395514824355863734,
0.981093777738395514824355863734, 2.20986485036028532578747169119, 2.83456106733617816587890482767, 3.96604206675158700563114268969, 4.72049182926131707319544056283, 5.65649319138181882137357518874, 6.28288998913628908072305703570, 7.38044722573046949148229558965, 7.82868470191142154949206770873, 8.661618020475539695421864554221
