L(s) = 1 | − 3-s + 5-s − 1.71·7-s + 9-s − 1.36·11-s + 0.754·13-s − 15-s − 5.06·17-s − 5.43·19-s + 1.71·21-s − 4.81·23-s + 25-s − 27-s + 9.51·29-s + 2.98·31-s + 1.36·33-s − 1.71·35-s − 3.95·37-s − 0.754·39-s − 4.98·41-s − 4.84·43-s + 45-s + 7.20·47-s − 4.04·49-s + 5.06·51-s + 3.05·53-s − 1.36·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.650·7-s + 0.333·9-s − 0.412·11-s + 0.209·13-s − 0.258·15-s − 1.22·17-s − 1.24·19-s + 0.375·21-s − 1.00·23-s + 0.200·25-s − 0.192·27-s + 1.76·29-s + 0.536·31-s + 0.238·33-s − 0.290·35-s − 0.650·37-s − 0.120·39-s − 0.778·41-s − 0.739·43-s + 0.149·45-s + 1.05·47-s − 0.577·49-s + 0.709·51-s + 0.419·53-s − 0.184·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.015284223\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.015284223\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
good | 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 13 | \( 1 - 0.754T + 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 + 4.81T + 23T^{2} \) |
| 29 | \( 1 - 9.51T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 + 4.98T + 41T^{2} \) |
| 43 | \( 1 + 4.84T + 43T^{2} \) |
| 47 | \( 1 - 7.20T + 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 5.80T + 61T^{2} \) |
| 71 | \( 1 - 2.01T + 71T^{2} \) |
| 73 | \( 1 - 7.75T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 1.49T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−7.88646114419767082224456537836, −6.73289419758876824671779085398, −6.52945624709687209101601274165, −5.93472565813304805526074163240, −4.97478393773296570466081141732, −4.45050035354877907011762032947, −3.55021492331590683242076740811, −2.54968595860358400937462276480, −1.84162736173756998742496848594, −0.49371402532621099290462706028,
0.49371402532621099290462706028, 1.84162736173756998742496848594, 2.54968595860358400937462276480, 3.55021492331590683242076740811, 4.45050035354877907011762032947, 4.97478393773296570466081141732, 5.93472565813304805526074163240, 6.52945624709687209101601274165, 6.73289419758876824671779085398, 7.88646114419767082224456537836