L(s) = 1 | + 2-s − 3-s + 4-s + 2.56·5-s − 6-s + 3.94·7-s + 8-s + 9-s + 2.56·10-s + 1.29·11-s − 12-s − 5.00·13-s + 3.94·14-s − 2.56·15-s + 16-s + 3.85·17-s + 18-s + 4.54·19-s + 2.56·20-s − 3.94·21-s + 1.29·22-s − 23-s − 24-s + 1.55·25-s − 5.00·26-s − 27-s + 3.94·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.14·5-s − 0.408·6-s + 1.48·7-s + 0.353·8-s + 0.333·9-s + 0.809·10-s + 0.391·11-s − 0.288·12-s − 1.38·13-s + 1.05·14-s − 0.661·15-s + 0.250·16-s + 0.936·17-s + 0.235·18-s + 1.04·19-s + 0.572·20-s − 0.860·21-s + 0.276·22-s − 0.208·23-s − 0.204·24-s + 0.311·25-s − 0.980·26-s − 0.192·27-s + 0.744·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.963856035\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.963856035\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 + 5.00T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 - 8.11T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 6.72T + 59T^{2} \) |
| 61 | \( 1 - 0.700T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 + 2.40T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−8.259761148335610486965996800732, −7.52940961751185737320280755065, −6.94458195706627106125820954514, −5.89612780913817228613341134790, −5.42074197584121960638570354391, −4.91217232238644652228323596588, −4.13417756365011973324090780155, −2.84761903455718244685414902469, −1.94211081450018309642751968232, −1.18204958808724015296986893410,
1.18204958808724015296986893410, 1.94211081450018309642751968232, 2.84761903455718244685414902469, 4.13417756365011973324090780155, 4.91217232238644652228323596588, 5.42074197584121960638570354391, 5.89612780913817228613341134790, 6.94458195706627106125820954514, 7.52940961751185737320280755065, 8.259761148335610486965996800732