| L(s) = 1 | + 0.499·2-s + 0.849·3-s − 1.75·4-s − 1.04·5-s + 0.424·6-s − 1.87·8-s − 2.27·9-s − 0.521·10-s + 3.96·11-s − 1.48·12-s − 0.885·15-s + 2.56·16-s − 0.142·17-s − 1.13·18-s + 5.50·19-s + 1.82·20-s + 1.98·22-s + 4.39·23-s − 1.59·24-s − 3.91·25-s − 4.48·27-s − 8.39·29-s − 0.442·30-s − 2.84·31-s + 5.03·32-s + 3.37·33-s − 0.0710·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.490·3-s − 0.875·4-s − 0.466·5-s + 0.173·6-s − 0.662·8-s − 0.759·9-s − 0.164·10-s + 1.19·11-s − 0.429·12-s − 0.228·15-s + 0.640·16-s − 0.0344·17-s − 0.268·18-s + 1.26·19-s + 0.407·20-s + 0.422·22-s + 0.915·23-s − 0.325·24-s − 0.782·25-s − 0.863·27-s − 1.55·29-s − 0.0808·30-s − 0.511·31-s + 0.889·32-s + 0.586·33-s − 0.0121·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707583912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707583912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.499T + 2T^{2} \) |
| 3 | \( 1 - 0.849T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 17 | \( 1 + 0.142T + 17T^{2} \) |
| 19 | \( 1 - 5.50T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 + 8.39T + 29T^{2} \) |
| 31 | \( 1 + 2.84T + 31T^{2} \) |
| 37 | \( 1 - 0.843T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 - 4.55T + 47T^{2} \) |
| 53 | \( 1 + 0.279T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 5.86T + 61T^{2} \) |
| 67 | \( 1 + 5.14T + 67T^{2} \) |
| 71 | \( 1 + 3.69T + 71T^{2} \) |
| 73 | \( 1 - 6.61T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 + 1.74T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−7.79443347818827400380762962164, −7.29403477474021111874383456538, −6.32241644728579084765850591439, −5.55772368851127023344855455302, −5.06649095664980752375747074563, −4.01986204151962106595220462204, −3.61791178323399445644545708511, −3.01012414483851747119663253855, −1.77523766906501670294428154673, −0.60049377029348509788900463433,
0.60049377029348509788900463433, 1.77523766906501670294428154673, 3.01012414483851747119663253855, 3.61791178323399445644545708511, 4.01986204151962106595220462204, 5.06649095664980752375747074563, 5.55772368851127023344855455302, 6.32241644728579084765850591439, 7.29403477474021111874383456538, 7.79443347818827400380762962164
