| L(s) = 1 | − 3-s − 0.0932·5-s + 3.86·7-s + 9-s + 4.61·11-s − 0.601·13-s + 0.0932·15-s − 0.832·17-s + 3.67·19-s − 3.86·21-s + 6.30·23-s − 4.99·25-s − 27-s + 4.62·29-s + 4.74·31-s − 4.61·33-s − 0.360·35-s + 5.70·37-s + 0.601·39-s − 4.02·41-s − 3.85·43-s − 0.0932·45-s + 2.09·47-s + 7.91·49-s + 0.832·51-s + 7.76·53-s − 0.430·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.0417·5-s + 1.45·7-s + 0.333·9-s + 1.39·11-s − 0.166·13-s + 0.0240·15-s − 0.201·17-s + 0.843·19-s − 0.842·21-s + 1.31·23-s − 0.998·25-s − 0.192·27-s + 0.858·29-s + 0.851·31-s − 0.803·33-s − 0.0608·35-s + 0.937·37-s + 0.0963·39-s − 0.627·41-s − 0.587·43-s − 0.0139·45-s + 0.305·47-s + 1.13·49-s + 0.116·51-s + 1.06·53-s − 0.0580·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.233397748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.233397748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.0932T + 5T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 + 0.601T + 13T^{2} \) |
| 17 | \( 1 + 0.832T + 17T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 - 4.62T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 + 3.85T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 + 4.38T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 + 8.12T + 79T^{2} \) |
| 83 | \( 1 + 2.46T + 83T^{2} \) |
| 89 | \( 1 - 4.23T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
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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
−8.459367290591264406102239245729, −7.64645538523445598274369903209, −7.00669541238048852040376945084, −6.22647860199013845874561875982, −5.40629394416984289391219155702, −4.65662748267384241833173327279, −4.14121116514762171113248276482, −2.94903573510515293466025988031, −1.66800482410551308773042223663, −0.989126221064150989092701187262,
0.989126221064150989092701187262, 1.66800482410551308773042223663, 2.94903573510515293466025988031, 4.14121116514762171113248276482, 4.65662748267384241833173327279, 5.40629394416984289391219155702, 6.22647860199013845874561875982, 7.00669541238048852040376945084, 7.64645538523445598274369903209, 8.459367290591264406102239245729
