L(s) = 1 | + 3·5-s + 4·11-s + 3·13-s − 4·17-s + 23-s + 4·25-s − 3·29-s − 6·31-s − 9·37-s + 9·41-s + 3·43-s + 7·47-s + 4·53-s + 12·55-s − 6·59-s − 10·61-s + 9·65-s − 4·67-s − 6·71-s + 8·73-s − 8·79-s − 4·83-s − 12·85-s − 14·89-s + 7·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 1.20·11-s + 0.832·13-s − 0.970·17-s + 0.208·23-s + 4/5·25-s − 0.557·29-s − 1.07·31-s − 1.47·37-s + 1.40·41-s + 0.457·43-s + 1.02·47-s + 0.549·53-s + 1.61·55-s − 0.781·59-s − 1.28·61-s + 1.11·65-s − 0.488·67-s − 0.712·71-s + 0.936·73-s − 0.900·79-s − 0.439·83-s − 1.30·85-s − 1.48·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.56496156497440, −13.12944461338540, −12.55359209207653, −12.23973558253352, −11.45983497185818, −11.11150438145898, −10.63215538917462, −10.24923441331726, −9.444281212287832, −9.227653598142821, −8.896972295314024, −8.431877287850043, −7.541148142512070, −7.099374446938770, −6.609090124647291, −6.022072172291713, −5.805240356865835, −5.239453067529919, −4.432433806187555, −4.040509457126481, −3.431131753472653, −2.714357572333653, −2.056291758261890, −1.578311719423415, −1.077917672493038, 0,
1.077917672493038, 1.578311719423415, 2.056291758261890, 2.714357572333653, 3.431131753472653, 4.040509457126481, 4.432433806187555, 5.239453067529919, 5.805240356865835, 6.022072172291713, 6.609090124647291, 7.099374446938770, 7.541148142512070, 8.431877287850043, 8.896972295314024, 9.227653598142821, 9.444281212287832, 10.24923441331726, 10.63215538917462, 11.11150438145898, 11.45983497185818, 12.23973558253352, 12.55359209207653, 13.12944461338540, 13.56496156497440