| L(s) = 1 | − 2.61·3-s − 2.61·5-s + 7-s + 3.82·9-s + 2.16·11-s − 0.448·13-s + 6.82·15-s − 7.65·17-s − 4.77·19-s − 2.61·21-s + 6.82·23-s + 1.82·25-s − 2.16·27-s − 9.55·29-s − 5.65·31-s − 5.65·33-s − 2.61·35-s − 5.22·37-s + 1.17·39-s + 3.65·41-s − 2.16·43-s − 10.0·45-s + 8·47-s + 49-s + 20.0·51-s + 10.4·53-s − 5.65·55-s + ⋯ |
| L(s) = 1 | − 1.50·3-s − 1.16·5-s + 0.377·7-s + 1.27·9-s + 0.652·11-s − 0.124·13-s + 1.76·15-s − 1.85·17-s − 1.09·19-s − 0.570·21-s + 1.42·23-s + 0.365·25-s − 0.416·27-s − 1.77·29-s − 1.01·31-s − 0.984·33-s − 0.441·35-s − 0.859·37-s + 0.187·39-s + 0.571·41-s − 0.330·43-s − 1.49·45-s + 1.16·47-s + 0.142·49-s + 2.80·51-s + 1.43·53-s − 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4910414288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4910414288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 + 0.448T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 + 9.55T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 0.448T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 3.06T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.079183569618779239239580969917, −8.644609258017325227073789010653, −7.31986265046127519764779240733, −6.99031986020231754436706468325, −6.07243382107648119891814640649, −5.16920629400447209468787383992, −4.38090758899624571929091877244, −3.77566626499754526914794114362, −2.03191357652435797897771359601, −0.50289367066627276125014389252,
0.50289367066627276125014389252, 2.03191357652435797897771359601, 3.77566626499754526914794114362, 4.38090758899624571929091877244, 5.16920629400447209468787383992, 6.07243382107648119891814640649, 6.99031986020231754436706468325, 7.31986265046127519764779240733, 8.644609258017325227073789010653, 9.079183569618779239239580969917
