| L(s) = 1 | + 3-s + 7-s − 2·9-s + 6·11-s − 5·13-s + 3·17-s + 21-s − 3·23-s − 5·25-s − 5·27-s − 9·29-s − 4·31-s + 6·33-s − 2·37-s − 5·39-s − 8·43-s − 6·49-s + 3·51-s + 3·53-s + 9·59-s − 10·61-s − 2·63-s + 5·67-s − 3·69-s − 6·71-s − 7·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.80·11-s − 1.38·13-s + 0.727·17-s + 0.218·21-s − 0.625·23-s − 25-s − 0.962·27-s − 1.67·29-s − 0.718·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s − 1.21·43-s − 6/7·49-s + 0.420·51-s + 0.412·53-s + 1.17·59-s − 1.28·61-s − 0.251·63-s + 0.610·67-s − 0.361·69-s − 0.712·71-s − 0.819·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.66995734298328287502507015405, −7.29904094327803276580864474565, −6.29607557013450001946456563759, −5.64317441971502851003917388776, −4.83778114166735308330677292657, −3.83932823582255789112395130179, −3.41239623655734813432880019975, −2.21851462150366027971745653477, −1.59049790592698824897087379644, 0,
1.59049790592698824897087379644, 2.21851462150366027971745653477, 3.41239623655734813432880019975, 3.83932823582255789112395130179, 4.83778114166735308330677292657, 5.64317441971502851003917388776, 6.29607557013450001946456563759, 7.29904094327803276580864474565, 7.66995734298328287502507015405
