L(s) = 1 | + 3-s + 0.324·5-s + 4.96·7-s + 9-s − 2.15·11-s + 2·13-s + 0.324·15-s + 5.02·17-s − 1.61·19-s + 4.96·21-s + 8·23-s − 4.89·25-s + 27-s + 2.31·29-s + 1.61·31-s − 2.15·33-s + 1.61·35-s − 6.96·37-s + 2·39-s + 9.86·41-s + 4.06·43-s + 0.324·45-s − 10.4·47-s + 17.6·49-s + 5.02·51-s − 4.89·53-s − 0.700·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.145·5-s + 1.87·7-s + 0.333·9-s − 0.650·11-s + 0.554·13-s + 0.0838·15-s + 1.21·17-s − 0.369·19-s + 1.08·21-s + 1.66·23-s − 0.978·25-s + 0.192·27-s + 0.429·29-s + 0.289·31-s − 0.375·33-s + 0.272·35-s − 1.14·37-s + 0.320·39-s + 1.54·41-s + 0.619·43-s + 0.0484·45-s − 1.52·47-s + 2.51·49-s + 0.703·51-s − 0.672·53-s − 0.0944·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.814808627\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.814808627\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 0.324T + 5T^{2} \) |
| 7 | \( 1 - 4.96T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 + 1.61T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 - 1.61T + 31T^{2} \) |
| 37 | \( 1 + 6.96T + 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 - 4.06T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.89T + 53T^{2} \) |
| 59 | \( 1 + 8.31T + 59T^{2} \) |
| 61 | \( 1 + 0.0303T + 61T^{2} \) |
| 67 | \( 1 - 6.71T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 - 4.26T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−7.85126566609432663734878137867, −7.49100698198828467698752625573, −6.49728512445957312250585750867, −5.53414944520892194423321004693, −5.04482415157639593257895431197, −4.37020740599241993160347859143, −3.46612934121373470598499432855, −2.61328585197223690846052515925, −1.73110281227054776356746292803, −1.03638863439279505037005614229,
1.03638863439279505037005614229, 1.73110281227054776356746292803, 2.61328585197223690846052515925, 3.46612934121373470598499432855, 4.37020740599241993160347859143, 5.04482415157639593257895431197, 5.53414944520892194423321004693, 6.49728512445957312250585750867, 7.49100698198828467698752625573, 7.85126566609432663734878137867