L(s) = 1 | + 2-s + 3.14·3-s + 4-s + 1.08·5-s + 3.14·6-s − 1.59·7-s + 8-s + 6.92·9-s + 1.08·10-s + 5.80·11-s + 3.14·12-s + 2.89·13-s − 1.59·14-s + 3.41·15-s + 16-s + 3.40·17-s + 6.92·18-s − 19-s + 1.08·20-s − 5.03·21-s + 5.80·22-s + 2.00·23-s + 3.14·24-s − 3.82·25-s + 2.89·26-s + 12.3·27-s − 1.59·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.81·3-s + 0.5·4-s + 0.484·5-s + 1.28·6-s − 0.604·7-s + 0.353·8-s + 2.30·9-s + 0.342·10-s + 1.75·11-s + 0.909·12-s + 0.803·13-s − 0.427·14-s + 0.880·15-s + 0.250·16-s + 0.825·17-s + 1.63·18-s − 0.229·19-s + 0.242·20-s − 1.09·21-s + 1.23·22-s + 0.417·23-s + 0.642·24-s − 0.765·25-s + 0.568·26-s + 2.37·27-s − 0.302·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.290737953\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.290737953\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 8.16T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 9.60T + 71T^{2} \) |
| 73 | \( 1 + 8.36T + 73T^{2} \) |
| 79 | \( 1 - 1.39T + 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 + 1.72T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.898941175812781832282927864029, −7.05273289109702920314003659619, −6.52948047969500401538016626506, −5.89977286365789278465431806965, −4.80573969725337167887236354353, −3.83443503302919644592968444384, −3.58423973585966244531872876205, −2.95378065611608403025426500936, −1.82483396257356583818491532127, −1.42562161583496808465957919754,
1.42562161583496808465957919754, 1.82483396257356583818491532127, 2.95378065611608403025426500936, 3.58423973585966244531872876205, 3.83443503302919644592968444384, 4.80573969725337167887236354353, 5.89977286365789278465431806965, 6.52948047969500401538016626506, 7.05273289109702920314003659619, 7.898941175812781832282927864029