| L(s) = 1 | + 3-s − 1.13·5-s − 3.14·7-s + 9-s + 3.49·11-s + 1.87·13-s − 1.13·15-s + 1.73·17-s − 2.67·19-s − 3.14·21-s − 6.17·23-s − 3.71·25-s + 27-s + 1.02·29-s + 6.27·31-s + 3.49·33-s + 3.57·35-s − 7.02·37-s + 1.87·39-s − 1.37·41-s + 9.09·43-s − 1.13·45-s − 8.48·47-s + 2.90·49-s + 1.73·51-s − 6.68·53-s − 3.97·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.507·5-s − 1.18·7-s + 0.333·9-s + 1.05·11-s + 0.521·13-s − 0.292·15-s + 0.420·17-s − 0.613·19-s − 0.686·21-s − 1.28·23-s − 0.742·25-s + 0.192·27-s + 0.189·29-s + 1.12·31-s + 0.609·33-s + 0.603·35-s − 1.15·37-s + 0.300·39-s − 0.214·41-s + 1.38·43-s − 0.169·45-s − 1.23·47-s + 0.415·49-s + 0.242·51-s − 0.918·53-s − 0.535·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 1.13T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 - 9.09T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 + 4.37T + 61T^{2} \) |
| 67 | \( 1 - 5.33T + 67T^{2} \) |
| 71 | \( 1 - 8.61T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 3.41T + 83T^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 + 5.22T + 97T^{2} \) |
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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
−7.996808521678481373460439852035, −6.85975157759615358043273902930, −6.46201763745502081598409360600, −5.79602602934902912188657172267, −4.56706117849189228572251683509, −3.78232842828488496445189984273, −3.46720350761965428541116483364, −2.42778040600765063491551611805, −1.35318720456044953306295906188, 0,
1.35318720456044953306295906188, 2.42778040600765063491551611805, 3.46720350761965428541116483364, 3.78232842828488496445189984273, 4.56706117849189228572251683509, 5.79602602934902912188657172267, 6.46201763745502081598409360600, 6.85975157759615358043273902930, 7.996808521678481373460439852035
