| L(s) = 1 | + 2-s + 4-s − 1.43·5-s − 1.87·7-s + 8-s − 1.43·10-s − 5.50·11-s + 5.81·13-s − 1.87·14-s + 16-s + 3.16·17-s − 0.890·19-s − 1.43·20-s − 5.50·22-s − 1.68·23-s − 2.94·25-s + 5.81·26-s − 1.87·28-s + 6.77·29-s + 2.18·31-s + 32-s + 3.16·34-s + 2.68·35-s + 9.53·37-s − 0.890·38-s − 1.43·40-s − 5.50·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.641·5-s − 0.707·7-s + 0.353·8-s − 0.453·10-s − 1.66·11-s + 1.61·13-s − 0.500·14-s + 0.250·16-s + 0.766·17-s − 0.204·19-s − 0.320·20-s − 1.17·22-s − 0.351·23-s − 0.588·25-s + 1.13·26-s − 0.353·28-s + 1.25·29-s + 0.392·31-s + 0.176·32-s + 0.542·34-s + 0.453·35-s + 1.56·37-s − 0.144·38-s − 0.226·40-s − 0.859·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 1.43T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 - 5.81T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 0.890T + 19T^{2} \) |
| 23 | \( 1 + 1.68T + 23T^{2} \) |
| 29 | \( 1 - 6.77T + 29T^{2} \) |
| 31 | \( 1 - 2.18T + 31T^{2} \) |
| 37 | \( 1 - 9.53T + 37T^{2} \) |
| 41 | \( 1 + 5.50T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + 9.50T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 0.993T + 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 3.41T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.071750791427634545464469122785, −7.40503633285576864599439032199, −6.26238013951809109167059377201, −6.04795519084846805760991546045, −4.97767703099070636403481815750, −4.28668423931572391692316515763, −3.23998008874760753875164178109, −2.97196068572264341363672821454, −1.51219697667423181271456436689, 0,
1.51219697667423181271456436689, 2.97196068572264341363672821454, 3.23998008874760753875164178109, 4.28668423931572391692316515763, 4.97767703099070636403481815750, 6.04795519084846805760991546045, 6.26238013951809109167059377201, 7.40503633285576864599439032199, 8.071750791427634545464469122785
