| L(s) = 1 | + 0.811·3-s − 0.636·5-s − 7-s − 2.34·9-s − 1.29·11-s − 1.02·13-s − 0.516·15-s − 7.09·17-s + 19-s − 0.811·21-s − 3.70·23-s − 4.59·25-s − 4.33·27-s − 4.94·29-s + 2.96·31-s − 1.05·33-s + 0.636·35-s − 2.78·37-s − 0.828·39-s + 5.64·41-s − 1.08·43-s + 1.48·45-s + 11.2·47-s + 49-s − 5.75·51-s + 6.23·53-s + 0.825·55-s + ⋯ |
| L(s) = 1 | + 0.468·3-s − 0.284·5-s − 0.377·7-s − 0.780·9-s − 0.391·11-s − 0.283·13-s − 0.133·15-s − 1.72·17-s + 0.229·19-s − 0.177·21-s − 0.771·23-s − 0.919·25-s − 0.834·27-s − 0.918·29-s + 0.532·31-s − 0.183·33-s + 0.107·35-s − 0.457·37-s − 0.132·39-s + 0.881·41-s − 0.165·43-s + 0.221·45-s + 1.63·47-s + 0.142·49-s − 0.806·51-s + 0.857·53-s + 0.111·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.033621924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033621924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.811T + 3T^{2} \) |
| 5 | \( 1 + 0.636T + 5T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 17 | \( 1 + 7.09T + 17T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 2.78T + 37T^{2} \) |
| 41 | \( 1 - 5.64T + 41T^{2} \) |
| 43 | \( 1 + 1.08T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 + 1.27T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 7.11T + 67T^{2} \) |
| 71 | \( 1 - 4.45T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 2.08T + 83T^{2} \) |
| 89 | \( 1 - 7.92T + 89T^{2} \) |
| 97 | \( 1 - 9.66T + 97T^{2} \) |
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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
−7.72961778662445013636968132176, −7.28824155661925908715011318098, −6.34073426624168775997540125686, −5.81139121391015738485781469393, −4.98463744833399596467434061899, −4.08362343646535925969837915930, −3.55130060072728447283973844501, −2.47727663257257431154858621512, −2.13789277956346894997400320308, −0.44959341818561389998374778731,
0.44959341818561389998374778731, 2.13789277956346894997400320308, 2.47727663257257431154858621512, 3.55130060072728447283973844501, 4.08362343646535925969837915930, 4.98463744833399596467434061899, 5.81139121391015738485781469393, 6.34073426624168775997540125686, 7.28824155661925908715011318098, 7.72961778662445013636968132176
