L(s) = 1 | + 2-s + 3-s + 4-s + 3.23·5-s + 6-s + 8-s + 9-s + 3.23·10-s − 0.763·11-s + 12-s + 3.23·13-s + 3.23·15-s + 16-s + 7.23·17-s + 18-s − 19-s + 3.23·20-s − 0.763·22-s + 0.763·23-s + 24-s + 5.47·25-s + 3.23·26-s + 27-s − 4.47·29-s + 3.23·30-s − 6.47·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.44·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.02·10-s − 0.230·11-s + 0.288·12-s + 0.897·13-s + 0.835·15-s + 0.250·16-s + 1.75·17-s + 0.235·18-s − 0.229·19-s + 0.723·20-s − 0.162·22-s + 0.159·23-s + 0.204·24-s + 1.09·25-s + 0.634·26-s + 0.192·27-s − 0.830·29-s + 0.590·30-s − 1.16·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.838855922\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.838855922\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 23 | \( 1 - 0.763T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 6.47T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 5.52T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 5.70T + 97T^{2} \) |
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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
−7.949125267703662448943301973953, −7.51341261841521734379512898341, −6.44464366500312635397871447642, −5.89989681469861216576395855619, −5.42160184223049131107804410887, −4.50782068479063759546497239644, −3.50274724647496148946517921456, −2.94178703665131840615315962462, −1.93550869446700512961426831069, −1.29436899002739379186805690245,
1.29436899002739379186805690245, 1.93550869446700512961426831069, 2.94178703665131840615315962462, 3.50274724647496148946517921456, 4.50782068479063759546497239644, 5.42160184223049131107804410887, 5.89989681469861216576395855619, 6.44464366500312635397871447642, 7.51341261841521734379512898341, 7.949125267703662448943301973953