| L(s) = 1 | − 1.19·2-s + 3-s − 0.579·4-s − 1.97·5-s − 1.19·6-s + 1.20·7-s + 3.07·8-s + 9-s + 2.35·10-s − 3.22·11-s − 0.579·12-s − 3.03·13-s − 1.43·14-s − 1.97·15-s − 2.50·16-s − 17-s − 1.19·18-s − 7.52·19-s + 1.14·20-s + 1.20·21-s + 3.84·22-s − 2.05·23-s + 3.07·24-s − 1.09·25-s + 3.61·26-s + 27-s − 0.699·28-s + ⋯ |
| L(s) = 1 | − 0.842·2-s + 0.577·3-s − 0.289·4-s − 0.883·5-s − 0.486·6-s + 0.456·7-s + 1.08·8-s + 0.333·9-s + 0.744·10-s − 0.973·11-s − 0.167·12-s − 0.841·13-s − 0.384·14-s − 0.509·15-s − 0.626·16-s − 0.242·17-s − 0.280·18-s − 1.72·19-s + 0.255·20-s + 0.263·21-s + 0.820·22-s − 0.429·23-s + 0.627·24-s − 0.219·25-s + 0.709·26-s + 0.192·27-s − 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6430735797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6430735797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 19 | \( 1 + 7.52T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 3.36T + 29T^{2} \) |
| 31 | \( 1 - 5.17T + 31T^{2} \) |
| 37 | \( 1 - 7.72T + 37T^{2} \) |
| 41 | \( 1 + 0.516T + 41T^{2} \) |
| 43 | \( 1 + 8.00T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 - 7.28T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 - 5.66T + 61T^{2} \) |
| 67 | \( 1 - 2.03T + 67T^{2} \) |
| 71 | \( 1 + 0.257T + 71T^{2} \) |
| 73 | \( 1 + 5.43T + 73T^{2} \) |
| 83 | \( 1 + 0.691T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.227680423230349664985052475950, −8.089370585328111527440150903445, −7.36810499074498009888857362135, −6.54323084038173519409200110183, −5.26160546659903931749585976234, −4.45109554479917461143808729118, −4.06376448035350906523260284540, −2.73497605388612765019619821162, −1.94265907258549080149516063555, −0.49332221966695619638419395166,
0.49332221966695619638419395166, 1.94265907258549080149516063555, 2.73497605388612765019619821162, 4.06376448035350906523260284540, 4.45109554479917461143808729118, 5.26160546659903931749585976234, 6.54323084038173519409200110183, 7.36810499074498009888857362135, 8.089370585328111527440150903445, 8.227680423230349664985052475950
