L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 6·11-s − 14-s + 16-s − 3·17-s − 19-s + 6·22-s − 3·23-s − 5·25-s + 28-s − 9·29-s + 4·31-s − 32-s + 3·34-s − 2·37-s + 38-s + 8·43-s − 6·44-s + 3·46-s − 6·49-s + 5·50-s + 3·53-s − 56-s + 9·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.80·11-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 1.27·22-s − 0.625·23-s − 25-s + 0.188·28-s − 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s − 0.328·37-s + 0.162·38-s + 1.21·43-s − 0.904·44-s + 0.442·46-s − 6/7·49-s + 0.707·50-s + 0.412·53-s − 0.133·56-s + 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1925745625\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1925745625\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.44502561150458, −13.72768503951752, −13.35823061914911, −12.82354529618561, −12.35293392526564, −11.60058466841159, −11.22662578845578, −10.74850295869575, −10.21827787400651, −9.824559020388022, −9.182780293495237, −8.587042447533767, −8.111204763545177, −7.604215910901854, −7.310019578811572, −6.488439078591862, −5.769150906190711, −5.504960372565398, −4.679713740302637, −4.118625896461153, −3.313522699771570, −2.490358592334484, −2.161145987298175, −1.368312020095128, −0.1637242386889945,
0.1637242386889945, 1.368312020095128, 2.161145987298175, 2.490358592334484, 3.313522699771570, 4.118625896461153, 4.679713740302637, 5.504960372565398, 5.769150906190711, 6.488439078591862, 7.310019578811572, 7.604215910901854, 8.111204763545177, 8.587042447533767, 9.182780293495237, 9.824559020388022, 10.21827787400651, 10.74850295869575, 11.22662578845578, 11.60058466841159, 12.35293392526564, 12.82354529618561, 13.35823061914911, 13.72768503951752, 14.44502561150458