L(s) = 1 | + 1.28·2-s − 2.01·3-s − 0.346·4-s − 2.59·6-s − 7-s − 3.01·8-s + 1.06·9-s − 11-s + 0.699·12-s + 1.52·13-s − 1.28·14-s − 3.18·16-s + 0.461·17-s + 1.37·18-s − 5.59·19-s + 2.01·21-s − 1.28·22-s + 5.33·23-s + 6.08·24-s + 1.95·26-s + 3.89·27-s + 0.346·28-s − 2.46·29-s − 3.92·31-s + 1.93·32-s + 2.01·33-s + 0.593·34-s + ⋯ |
L(s) = 1 | + 0.909·2-s − 1.16·3-s − 0.173·4-s − 1.05·6-s − 0.377·7-s − 1.06·8-s + 0.356·9-s − 0.301·11-s + 0.201·12-s + 0.422·13-s − 0.343·14-s − 0.796·16-s + 0.112·17-s + 0.324·18-s − 1.28·19-s + 0.440·21-s − 0.274·22-s + 1.11·23-s + 1.24·24-s + 0.383·26-s + 0.749·27-s + 0.0655·28-s − 0.457·29-s − 0.705·31-s + 0.342·32-s + 0.351·33-s + 0.101·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.056567148\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.056567148\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.28T + 2T^{2} \) |
| 3 | \( 1 + 2.01T + 3T^{2} \) |
| 13 | \( 1 - 1.52T + 13T^{2} \) |
| 17 | \( 1 - 0.461T + 17T^{2} \) |
| 19 | \( 1 + 5.59T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 3.92T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 - 4.95T + 41T^{2} \) |
| 43 | \( 1 - 8.80T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 + 0.386T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 - 8.97T + 61T^{2} \) |
| 67 | \( 1 - 7.72T + 67T^{2} \) |
| 71 | \( 1 - 3.38T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 7.16T + 79T^{2} \) |
| 83 | \( 1 - 9.78T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−9.063920067882503184301370258301, −8.645266382347989420714318634336, −7.28934275055608185571563005559, −6.50748712223904531929844348388, −5.75148480693859237353186855118, −5.32310042433753125546100436079, −4.38641905605778099809515008755, −3.61964417242566639048810212929, −2.47455102973592910919723627092, −0.62021219543234726263114444114,
0.62021219543234726263114444114, 2.47455102973592910919723627092, 3.61964417242566639048810212929, 4.38641905605778099809515008755, 5.32310042433753125546100436079, 5.75148480693859237353186855118, 6.50748712223904531929844348388, 7.28934275055608185571563005559, 8.645266382347989420714318634336, 9.063920067882503184301370258301