| L(s) = 1 | − 2-s + 4-s − 2.56·5-s + 2.27·7-s − 8-s + 2.56·10-s + 0.680·11-s + 3.64·13-s − 2.27·14-s + 16-s + 1.93·17-s − 1.65·19-s − 2.56·20-s − 0.680·22-s − 6.35·23-s + 1.57·25-s − 3.64·26-s + 2.27·28-s − 8.02·29-s − 0.995·31-s − 32-s − 1.93·34-s − 5.84·35-s − 8.32·37-s + 1.65·38-s + 2.56·40-s + 2.25·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.14·5-s + 0.861·7-s − 0.353·8-s + 0.811·10-s + 0.205·11-s + 1.01·13-s − 0.609·14-s + 0.250·16-s + 0.470·17-s − 0.380·19-s − 0.573·20-s − 0.145·22-s − 1.32·23-s + 0.315·25-s − 0.714·26-s + 0.430·28-s − 1.49·29-s − 0.178·31-s − 0.176·32-s − 0.332·34-s − 0.988·35-s − 1.36·37-s + 0.269·38-s + 0.405·40-s + 0.352·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 11 | \( 1 - 0.680T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 + 6.35T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 + 0.995T + 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 - 2.25T + 41T^{2} \) |
| 43 | \( 1 - 0.273T + 43T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 - 7.97T + 53T^{2} \) |
| 59 | \( 1 - 3.85T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 + 2.15T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 1.47T + 83T^{2} \) |
| 89 | \( 1 + 18.7T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−8.268060846074636608516634095277, −7.55058883805611044672758893452, −6.87121832850627351869156747042, −5.91964554323610400349854951642, −5.14671560629008302062102785098, −3.87452469134480289589124007626, −3.73230205945844464290148775015, −2.22054872621893301435555082973, −1.30350023656048769239704007898, 0,
1.30350023656048769239704007898, 2.22054872621893301435555082973, 3.73230205945844464290148775015, 3.87452469134480289589124007626, 5.14671560629008302062102785098, 5.91964554323610400349854951642, 6.87121832850627351869156747042, 7.55058883805611044672758893452, 8.268060846074636608516634095277
