| L(s) = 1 | + 3-s − 2.01·5-s − 2.53·7-s + 9-s + 4.24·11-s − 2.95·13-s − 2.01·15-s + 7.86·17-s − 7.02·19-s − 2.53·21-s − 2.38·23-s − 0.921·25-s + 27-s − 0.629·29-s + 4.91·31-s + 4.24·33-s + 5.11·35-s + 1.38·37-s − 2.95·39-s + 4.74·41-s − 1.92·43-s − 2.01·45-s + 4.57·47-s − 0.586·49-s + 7.86·51-s − 5.90·53-s − 8.57·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.903·5-s − 0.957·7-s + 0.333·9-s + 1.27·11-s − 0.818·13-s − 0.521·15-s + 1.90·17-s − 1.61·19-s − 0.552·21-s − 0.496·23-s − 0.184·25-s + 0.192·27-s − 0.116·29-s + 0.883·31-s + 0.738·33-s + 0.864·35-s + 0.227·37-s − 0.472·39-s + 0.740·41-s − 0.292·43-s − 0.301·45-s + 0.667·47-s − 0.0837·49-s + 1.10·51-s − 0.810·53-s − 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 7.86T + 17T^{2} \) |
| 19 | \( 1 + 7.02T + 19T^{2} \) |
| 23 | \( 1 + 2.38T + 23T^{2} \) |
| 29 | \( 1 + 0.629T + 29T^{2} \) |
| 31 | \( 1 - 4.91T + 31T^{2} \) |
| 37 | \( 1 - 1.38T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 + 1.92T + 43T^{2} \) |
| 47 | \( 1 - 4.57T + 47T^{2} \) |
| 53 | \( 1 + 5.90T + 53T^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 5.20T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 5.38T + 73T^{2} \) |
| 79 | \( 1 - 8.24T + 79T^{2} \) |
| 83 | \( 1 + 3.55T + 83T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−7.84968692252616779737942319246, −7.05764734746185259340095819694, −6.43126188909939301024839959058, −5.72093543724890486620360983648, −4.49510547536301526853655144324, −3.95672299548191693129077558326, −3.33315044520020396324100100670, −2.50720200210585852112573438092, −1.29125001879399607322396531797, 0,
1.29125001879399607322396531797, 2.50720200210585852112573438092, 3.33315044520020396324100100670, 3.95672299548191693129077558326, 4.49510547536301526853655144324, 5.72093543724890486620360983648, 6.43126188909939301024839959058, 7.05764734746185259340095819694, 7.84968692252616779737942319246
