| L(s) = 1 | + 1.30·3-s − 1.69·5-s − 3.93·7-s − 1.30·9-s + 3.01·11-s + 6.58·13-s − 2.20·15-s + 2.00·17-s − 5.92·19-s − 5.11·21-s − 5.10·23-s − 2.14·25-s − 5.60·27-s + 3.88·29-s + 8.26·31-s + 3.92·33-s + 6.64·35-s − 3.68·37-s + 8.56·39-s + 1.89·41-s − 7.00·43-s + 2.20·45-s + 9.72·47-s + 8.46·49-s + 2.61·51-s + 6.65·53-s − 5.09·55-s + ⋯ |
| L(s) = 1 | + 0.751·3-s − 0.756·5-s − 1.48·7-s − 0.435·9-s + 0.909·11-s + 1.82·13-s − 0.568·15-s + 0.486·17-s − 1.35·19-s − 1.11·21-s − 1.06·23-s − 0.428·25-s − 1.07·27-s + 0.722·29-s + 1.48·31-s + 0.683·33-s + 1.12·35-s − 0.605·37-s + 1.37·39-s + 0.296·41-s − 1.06·43-s + 0.328·45-s + 1.41·47-s + 1.20·49-s + 0.365·51-s + 0.913·53-s − 0.687·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630257793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630257793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 + 5.92T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 - 1.89T + 41T^{2} \) |
| 43 | \( 1 + 7.00T + 43T^{2} \) |
| 47 | \( 1 - 9.72T + 47T^{2} \) |
| 53 | \( 1 - 6.65T + 53T^{2} \) |
| 59 | \( 1 + 9.91T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 2.42T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 6.40T + 79T^{2} \) |
| 83 | \( 1 - 6.22T + 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 - 5.85T + 97T^{2} \) |
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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
−8.360408503098824748335568970079, −8.082042326436436083556321371392, −6.81097146232143205336202009296, −6.34257365178394989624015261707, −5.77108104065306399995414773251, −4.19583926605701757558864183268, −3.72883695951431140265635637453, −3.21769973487256896608207161541, −2.14073618714155450844060358532, −0.68891329995922413371368203372,
0.68891329995922413371368203372, 2.14073618714155450844060358532, 3.21769973487256896608207161541, 3.72883695951431140265635637453, 4.19583926605701757558864183268, 5.77108104065306399995414773251, 6.34257365178394989624015261707, 6.81097146232143205336202009296, 8.082042326436436083556321371392, 8.360408503098824748335568970079
