| L(s) = 1 | + 1.73·3-s + 2.27·5-s + 7-s + 0.0242·9-s − 4.95·11-s − 4.44·13-s + 3.95·15-s + 7.01·17-s − 3.83·19-s + 1.73·21-s − 3.65·23-s + 0.162·25-s − 5.17·27-s − 4.08·29-s − 1.91·31-s − 8.62·33-s + 2.27·35-s − 5.86·37-s − 7.73·39-s + 8.15·41-s − 8.27·43-s + 0.0550·45-s + 4.09·47-s + 49-s + 12.2·51-s + 7.42·53-s − 11.2·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 1.01·5-s + 0.377·7-s + 0.00807·9-s − 1.49·11-s − 1.23·13-s + 1.02·15-s + 1.70·17-s − 0.880·19-s + 0.379·21-s − 0.762·23-s + 0.0324·25-s − 0.995·27-s − 0.757·29-s − 0.344·31-s − 1.50·33-s + 0.384·35-s − 0.963·37-s − 1.23·39-s + 1.27·41-s − 1.26·43-s + 0.00820·45-s + 0.597·47-s + 0.142·49-s + 1.70·51-s + 1.01·53-s − 1.51·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 11 | \( 1 + 4.95T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 - 7.01T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + 1.91T + 31T^{2} \) |
| 37 | \( 1 + 5.86T + 37T^{2} \) |
| 41 | \( 1 - 8.15T + 41T^{2} \) |
| 43 | \( 1 + 8.27T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 - 7.42T + 53T^{2} \) |
| 59 | \( 1 - 5.57T + 59T^{2} \) |
| 61 | \( 1 + 9.05T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 6.19T + 79T^{2} \) |
| 83 | \( 1 + 5.11T + 83T^{2} \) |
| 89 | \( 1 + 7.33T + 89T^{2} \) |
| 97 | \( 1 + 7.10T + 97T^{2} \) |
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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
−7.64664053131547330208540290205, −7.22179342982317329650593949850, −5.85366993800029333048723270687, −5.60142323784213643238939281688, −4.84793493346998757056503689806, −3.80563086090737759318522223258, −2.88669960244637617672288376341, −2.33777118306840321834492769319, −1.70414060720933370828825573229, 0,
1.70414060720933370828825573229, 2.33777118306840321834492769319, 2.88669960244637617672288376341, 3.80563086090737759318522223258, 4.84793493346998757056503689806, 5.60142323784213643238939281688, 5.85366993800029333048723270687, 7.22179342982317329650593949850, 7.64664053131547330208540290205
