| L(s) = 1 | − 1.41·3-s + 7-s − 0.999·9-s + 2.82·11-s + 2.82·13-s + 6·17-s − 4.24·19-s − 1.41·21-s − 5·25-s + 5.65·27-s − 1.41·29-s − 10·31-s − 4.00·33-s − 4.24·37-s − 4.00·39-s − 10·41-s − 8.48·43-s + 2·47-s + 49-s − 8.48·51-s + 12.7·53-s + 6·57-s − 1.41·59-s − 0.999·63-s + 11.3·67-s − 12·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.816·3-s + 0.377·7-s − 0.333·9-s + 0.852·11-s + 0.784·13-s + 1.45·17-s − 0.973·19-s − 0.308·21-s − 25-s + 1.08·27-s − 0.262·29-s − 1.79·31-s − 0.696·33-s − 0.697·37-s − 0.640·39-s − 1.56·41-s − 1.29·43-s + 0.291·47-s + 0.142·49-s − 1.18·51-s + 1.74·53-s + 0.794·57-s − 0.184·59-s − 0.125·63-s + 1.38·67-s − 1.42·71-s + 0.234·73-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 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.46473132756715875357590270316, −6.80882354276778293913162940303, −6.02494145781311797519071895620, −5.58427485792964587314152582088, −4.92092142676527129987847930432, −3.83208359418711170321037230257, −3.43957009147269527585473042151, −2.03573406855574972793299962833, −1.24047120954098137707754470380, 0,
1.24047120954098137707754470380, 2.03573406855574972793299962833, 3.43957009147269527585473042151, 3.83208359418711170321037230257, 4.92092142676527129987847930432, 5.58427485792964587314152582088, 6.02494145781311797519071895620, 6.80882354276778293913162940303, 7.46473132756715875357590270316
