L(s) = 1 | − 4·2-s − 26.3·3-s + 16·4-s + 84.5·5-s + 105.·6-s + 15.2·7-s − 64·8-s + 452.·9-s − 338.·10-s − 450.·11-s − 421.·12-s − 802.·13-s − 61.0·14-s − 2.22e3·15-s + 256·16-s − 1.33e3·17-s − 1.80e3·18-s + 1.35e3·20-s − 402.·21-s + 1.80e3·22-s + 379.·23-s + 1.68e3·24-s + 4.01e3·25-s + 3.20e3·26-s − 5.51e3·27-s + 244.·28-s − 8.33e3·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.69·3-s + 0.5·4-s + 1.51·5-s + 1.19·6-s + 0.117·7-s − 0.353·8-s + 1.86·9-s − 1.06·10-s − 1.12·11-s − 0.845·12-s − 1.31·13-s − 0.0832·14-s − 2.55·15-s + 0.250·16-s − 1.11·17-s − 1.31·18-s + 0.755·20-s − 0.199·21-s + 0.793·22-s + 0.149·23-s + 0.597·24-s + 1.28·25-s + 0.931·26-s − 1.45·27-s + 0.0588·28-s − 1.83·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5124857175\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5124857175\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 26.3T + 243T^{2} \) |
| 5 | \( 1 - 84.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 15.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 450.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 802.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.33e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 379.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.25e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.47e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.23e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.64e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.53e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.36e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.07e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.14e5T + 8.58e9T^{2} \) |
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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
−9.935218381044817429580020588794, −9.124246797589859504764833183190, −7.73100808279993128678628087782, −6.83405827892360782041582076567, −6.10043616637002083292788204780, −5.33770723246464569089662566993, −4.71686116302366778772696991962, −2.53826932594049851730725767910, −1.70965307044532875971766796508, −0.39616416081856741281837230991,
0.39616416081856741281837230991, 1.70965307044532875971766796508, 2.53826932594049851730725767910, 4.71686116302366778772696991962, 5.33770723246464569089662566993, 6.10043616637002083292788204780, 6.83405827892360782041582076567, 7.73100808279993128678628087782, 9.124246797589859504764833183190, 9.935218381044817429580020588794