| L(s) = 1 | + 1.14·3-s − 2.02·5-s + 3.81·7-s − 1.69·9-s − 1.49·11-s + 3.70·13-s − 2.31·15-s + 4.98·17-s + 2.85·19-s + 4.35·21-s − 6.43·23-s − 0.881·25-s − 5.35·27-s − 3.64·29-s + 2.05·31-s − 1.70·33-s − 7.74·35-s + 10.4·37-s + 4.22·39-s + 9.36·41-s + 11.0·43-s + 3.44·45-s − 6.32·47-s + 7.56·49-s + 5.68·51-s + 4.92·53-s + 3.03·55-s + ⋯ |
| L(s) = 1 | + 0.658·3-s − 0.907·5-s + 1.44·7-s − 0.566·9-s − 0.451·11-s + 1.02·13-s − 0.597·15-s + 1.20·17-s + 0.655·19-s + 0.949·21-s − 1.34·23-s − 0.176·25-s − 1.03·27-s − 0.676·29-s + 0.368·31-s − 0.297·33-s − 1.30·35-s + 1.71·37-s + 0.676·39-s + 1.46·41-s + 1.68·43-s + 0.514·45-s − 0.922·47-s + 1.08·49-s + 0.795·51-s + 0.676·53-s + 0.409·55-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
| \(L(1)\) |
\(\approx\) |
\(2.394566162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394566162\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 23 | \( 1 + 6.43T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 - 2.05T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 9.36T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 - 4.92T + 53T^{2} \) |
| 59 | \( 1 + 0.897T + 59T^{2} \) |
| 61 | \( 1 + 0.628T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 6.59T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 0.00831T + 97T^{2} \) |
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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
−8.118501260728094180060117564534, −7.85618901502940241349147726254, −7.55543221087807672181072957307, −5.97696978765019914180779618381, −5.59205139083180708980849410791, −4.45461986943142611629105518073, −3.86809563436718981008034212013, −3.01654928465193020070506473167, −2.02664015613404937378590669267, −0.886136011155106412219023216199,
0.886136011155106412219023216199, 2.02664015613404937378590669267, 3.01654928465193020070506473167, 3.86809563436718981008034212013, 4.45461986943142611629105518073, 5.59205139083180708980849410791, 5.97696978765019914180779618381, 7.55543221087807672181072957307, 7.85618901502940241349147726254, 8.118501260728094180060117564534
