| L(s) = 1 | + 2.95·3-s − 1.95·5-s + 5.73·9-s − 2.65·11-s + 1.50·13-s − 5.77·15-s − 2.09·17-s − 6.38·19-s + 2.12·23-s − 1.18·25-s + 8.07·27-s − 7.42·29-s − 0.0654·31-s − 7.83·33-s + 0.537·37-s + 4.43·39-s − 11.8·41-s + 0.588·43-s − 11.1·45-s + 3.02·47-s − 6.18·51-s + 1.04·53-s + 5.17·55-s − 18.8·57-s − 2.27·59-s + 10.2·61-s − 2.93·65-s + ⋯ |
| L(s) = 1 | + 1.70·3-s − 0.873·5-s + 1.91·9-s − 0.799·11-s + 0.416·13-s − 1.48·15-s − 0.507·17-s − 1.46·19-s + 0.442·23-s − 0.237·25-s + 1.55·27-s − 1.37·29-s − 0.0117·31-s − 1.36·33-s + 0.0884·37-s + 0.710·39-s − 1.85·41-s + 0.0897·43-s − 1.66·45-s + 0.441·47-s − 0.865·51-s + 0.143·53-s + 0.698·55-s − 2.50·57-s − 0.295·59-s + 1.31·61-s − 0.363·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.95T + 3T^{2} \) |
| 5 | \( 1 + 1.95T + 5T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 - 2.12T + 23T^{2} \) |
| 29 | \( 1 + 7.42T + 29T^{2} \) |
| 31 | \( 1 + 0.0654T + 31T^{2} \) |
| 37 | \( 1 - 0.537T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 0.588T + 43T^{2} \) |
| 47 | \( 1 - 3.02T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 + 2.27T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 0.221T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 3.51T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.964363659537588151874181814534, −7.32853185145782981344292762654, −6.70581970258867667306620305973, −5.58282893637031744896509781202, −4.50427196503136681231061103925, −3.93805340381470038492693525728, −3.29032899599354060052231850915, −2.44974649897763804535483498043, −1.70792936231248930872098194297, 0,
1.70792936231248930872098194297, 2.44974649897763804535483498043, 3.29032899599354060052231850915, 3.93805340381470038492693525728, 4.50427196503136681231061103925, 5.58282893637031744896509781202, 6.70581970258867667306620305973, 7.32853185145782981344292762654, 7.964363659537588151874181814534
