| L(s) = 1 | − 1.48·2-s + 3-s + 0.214·4-s − 3.58·5-s − 1.48·6-s + 0.0236·7-s + 2.65·8-s + 9-s + 5.33·10-s − 3.71·11-s + 0.214·12-s − 13-s − 0.0351·14-s − 3.58·15-s − 4.38·16-s + 2.38·17-s − 1.48·18-s + 1.67·19-s − 0.769·20-s + 0.0236·21-s + 5.52·22-s − 5.50·23-s + 2.65·24-s + 7.86·25-s + 1.48·26-s + 27-s + 0.00507·28-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 0.577·3-s + 0.107·4-s − 1.60·5-s − 0.607·6-s + 0.00893·7-s + 0.939·8-s + 0.333·9-s + 1.68·10-s − 1.12·11-s + 0.0619·12-s − 0.277·13-s − 0.00940·14-s − 0.926·15-s − 1.09·16-s + 0.578·17-s − 0.350·18-s + 0.384·19-s − 0.172·20-s + 0.00516·21-s + 1.17·22-s − 1.14·23-s + 0.542·24-s + 1.57·25-s + 0.291·26-s + 0.192·27-s + 0.000958·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4376430652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4376430652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 7 | \( 1 - 0.0236T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 + 5.50T + 23T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 - 2.89T + 31T^{2} \) |
| 37 | \( 1 + 3.51T + 37T^{2} \) |
| 41 | \( 1 + 6.74T + 41T^{2} \) |
| 43 | \( 1 + 0.758T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 - 7.86T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 67 | \( 1 + 9.67T + 67T^{2} \) |
| 71 | \( 1 + 2.86T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 4.59T + 83T^{2} \) |
| 89 | \( 1 - 9.99T + 89T^{2} \) |
| 97 | \( 1 + 3.56T + 97T^{2} \) |
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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
−8.160653991384947457074638972357, −7.959411143319043870276964498893, −7.51312092818967756410000520892, −6.66407705318773258865750883375, −5.26310409445060542738050958159, −4.57362317011924594899484709819, −3.74534622980545344046924872307, −2.98791605107813720189039007367, −1.76491369187973638388144685125, −0.42795546560926930904963166012,
0.42795546560926930904963166012, 1.76491369187973638388144685125, 2.98791605107813720189039007367, 3.74534622980545344046924872307, 4.57362317011924594899484709819, 5.26310409445060542738050958159, 6.66407705318773258865750883375, 7.51312092818967756410000520892, 7.959411143319043870276964498893, 8.160653991384947457074638972357
