L(s) = 1 | + 3·5-s − 3·9-s − 2·11-s − 4·13-s − 19-s − 4·23-s + 4·25-s + 6·29-s + 10·37-s − 7·41-s + 10·43-s − 9·45-s − 12·47-s + 4·53-s − 6·55-s − 3·59-s + 12·61-s − 12·65-s − 12·67-s − 13·71-s + 2·73-s − 6·79-s + 9·81-s + 6·83-s − 10·89-s − 3·95-s − 97-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 9-s − 0.603·11-s − 1.10·13-s − 0.229·19-s − 0.834·23-s + 4/5·25-s + 1.11·29-s + 1.64·37-s − 1.09·41-s + 1.52·43-s − 1.34·45-s − 1.75·47-s + 0.549·53-s − 0.809·55-s − 0.390·59-s + 1.53·61-s − 1.48·65-s − 1.46·67-s − 1.54·71-s + 0.234·73-s − 0.675·79-s + 81-s + 0.658·83-s − 1.05·89-s − 0.307·95-s − 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.435587779\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.435587779\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.51975528209943, −11.96754347714444, −11.67949737285591, −11.11057685257341, −10.55791691379925, −10.05355527981275, −9.956077809651248, −9.372409229260841, −8.881750952795836, −8.469643965523684, −7.825194340664856, −7.629314716931248, −6.787732960282320, −6.388667915644991, −5.987858005326486, −5.456874035654483, −5.193363757351086, −4.537557397133286, −4.110638794325769, −3.119803234025569, −2.787861974692893, −2.354434853581591, −1.888251936421188, −1.141705931015917, −0.3025419876485364,
0.3025419876485364, 1.141705931015917, 1.888251936421188, 2.354434853581591, 2.787861974692893, 3.119803234025569, 4.110638794325769, 4.537557397133286, 5.193363757351086, 5.456874035654483, 5.987858005326486, 6.388667915644991, 6.787732960282320, 7.629314716931248, 7.825194340664856, 8.469643965523684, 8.881750952795836, 9.372409229260841, 9.956077809651248, 10.05355527981275, 10.55791691379925, 11.11057685257341, 11.67949737285591, 11.96754347714444, 12.51975528209943