| L(s) = 1 | − 5·7-s + 11-s − 13-s − 3·17-s − 6·19-s + 7·23-s − 6·29-s + 2·31-s − 37-s − 7·41-s + 8·43-s − 2·47-s + 18·49-s + 13·53-s − 8·59-s − 7·61-s + 12·67-s + 71-s + 10·73-s − 5·77-s − 3·79-s − 8·83-s − 13·89-s + 5·91-s − 7·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 0.301·11-s − 0.277·13-s − 0.727·17-s − 1.37·19-s + 1.45·23-s − 1.11·29-s + 0.359·31-s − 0.164·37-s − 1.09·41-s + 1.21·43-s − 0.291·47-s + 18/7·49-s + 1.78·53-s − 1.04·59-s − 0.896·61-s + 1.46·67-s + 0.118·71-s + 1.17·73-s − 0.569·77-s − 0.337·79-s − 0.878·83-s − 1.37·89-s + 0.524·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5719051534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5719051534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.90975553456112, −13.88199499078302, −13.49321092390972, −13.04877748142736, −12.50247456353213, −12.34909410585243, −11.43017528084470, −10.88021747133657, −10.50721873918392, −9.715072487211490, −9.501868934637533, −8.794549856206188, −8.557674186884738, −7.524218851928767, −6.955289657149566, −6.644296641371498, −6.119121696744682, −5.485120277656157, −4.758684978345879, −3.938156379837562, −3.656690887945405, −2.712548668862920, −2.429350434669438, −1.315192294027169, −0.2718590986394657,
0.2718590986394657, 1.315192294027169, 2.429350434669438, 2.712548668862920, 3.656690887945405, 3.938156379837562, 4.758684978345879, 5.485120277656157, 6.119121696744682, 6.644296641371498, 6.955289657149566, 7.524218851928767, 8.557674186884738, 8.794549856206188, 9.501868934637533, 9.715072487211490, 10.50721873918392, 10.88021747133657, 11.43017528084470, 12.34909410585243, 12.50247456353213, 13.04877748142736, 13.49321092390972, 13.88199499078302, 14.90975553456112
