| L(s) = 1 | − 8.11·3-s + 11.0·5-s + 5.74·7-s + 38.9·9-s − 34.5·11-s − 29.8·13-s − 89.5·15-s − 17·17-s − 115.·19-s − 46.6·21-s − 84.9·23-s − 3.34·25-s − 96.8·27-s − 144.·29-s − 24.1·31-s + 280.·33-s + 63.3·35-s − 221.·37-s + 242.·39-s + 345.·41-s + 540.·43-s + 429.·45-s − 354.·47-s − 309.·49-s + 138.·51-s + 66.6·53-s − 380.·55-s + ⋯ |
| L(s) = 1 | − 1.56·3-s + 0.986·5-s + 0.310·7-s + 1.44·9-s − 0.946·11-s − 0.637·13-s − 1.54·15-s − 0.242·17-s − 1.39·19-s − 0.484·21-s − 0.769·23-s − 0.0267·25-s − 0.690·27-s − 0.923·29-s − 0.139·31-s + 1.47·33-s + 0.306·35-s − 0.982·37-s + 0.995·39-s + 1.31·41-s + 1.91·43-s + 1.42·45-s − 1.10·47-s − 0.903·49-s + 0.378·51-s + 0.172·53-s − 0.933·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 8.11T + 27T^{2} \) |
| 5 | \( 1 - 11.0T + 125T^{2} \) |
| 7 | \( 1 - 5.74T + 343T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29.8T + 2.19e3T^{2} \) |
| 19 | \( 1 + 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 24.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 540.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 66.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 611.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 623.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 566.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 937.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 707.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 306.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 707.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−12.29094270147578961734211407234, −11.01408598733745475455570490440, −10.48701517103157721976372895781, −9.397940438322630446983581752264, −7.73184888265171993177669768022, −6.35941399066934966724500898673, −5.60465670304312586916904788490, −4.59325531001353832336264250029, −2.04769079423803862635813323333, 0,
2.04769079423803862635813323333, 4.59325531001353832336264250029, 5.60465670304312586916904788490, 6.35941399066934966724500898673, 7.73184888265171993177669768022, 9.397940438322630446983581752264, 10.48701517103157721976372895781, 11.01408598733745475455570490440, 12.29094270147578961734211407234
