| L(s) = 1 | + 1.60·3-s + 2.61·5-s − 1.65·7-s − 0.432·9-s + 0.445·11-s − 1.68·13-s + 4.18·15-s − 3.40·17-s − 0.277·19-s − 2.64·21-s + 0.377·23-s + 1.81·25-s − 5.50·27-s − 4.89·29-s − 7.98·31-s + 0.713·33-s − 4.30·35-s + 0.844·37-s − 2.69·39-s − 6.97·41-s − 3.08·43-s − 1.12·45-s + 6.48·47-s − 4.27·49-s − 5.44·51-s − 4.55·53-s + 1.16·55-s + ⋯ |
| L(s) = 1 | + 0.925·3-s + 1.16·5-s − 0.623·7-s − 0.144·9-s + 0.134·11-s − 0.465·13-s + 1.07·15-s − 0.824·17-s − 0.0635·19-s − 0.576·21-s + 0.0787·23-s + 0.362·25-s − 1.05·27-s − 0.908·29-s − 1.43·31-s + 0.124·33-s − 0.728·35-s + 0.138·37-s − 0.431·39-s − 1.08·41-s − 0.470·43-s − 0.168·45-s + 0.945·47-s − 0.611·49-s − 0.762·51-s − 0.625·53-s + 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 1.60T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 - 0.445T + 11T^{2} \) |
| 13 | \( 1 + 1.68T + 13T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 + 0.277T + 19T^{2} \) |
| 23 | \( 1 - 0.377T + 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 + 7.98T + 31T^{2} \) |
| 37 | \( 1 - 0.844T + 37T^{2} \) |
| 41 | \( 1 + 6.97T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 4.55T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 9.80T + 67T^{2} \) |
| 71 | \( 1 - 0.193T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 9.29T + 83T^{2} \) |
| 89 | \( 1 + 3.44T + 89T^{2} \) |
| 97 | \( 1 - 8.56T + 97T^{2} \) |
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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
−7.72556653951570883998142229979, −7.05081905502948827916017148013, −6.27443174701542294201245040293, −5.67933457408732644213297565160, −4.90183074375338772808492346012, −3.81336422461875659259108938792, −3.13558308867264244253242117898, −2.26857978726888189392117583447, −1.75264866163212812719400651959, 0,
1.75264866163212812719400651959, 2.26857978726888189392117583447, 3.13558308867264244253242117898, 3.81336422461875659259108938792, 4.90183074375338772808492346012, 5.67933457408732644213297565160, 6.27443174701542294201245040293, 7.05081905502948827916017148013, 7.72556653951570883998142229979
