| L(s) = 1 | + 3-s − 2·9-s + 2·11-s + 2·17-s − 8·19-s − 23-s − 5·27-s + 9·29-s + 4·31-s + 2·33-s − 6·37-s + 2·41-s + 9·43-s − 4·47-s − 7·49-s + 2·51-s + 13·53-s − 8·57-s − 6·59-s − 5·61-s − 6·67-s − 69-s − 2·71-s − 8·73-s + 17·79-s + 81-s + 6·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.603·11-s + 0.485·17-s − 1.83·19-s − 0.208·23-s − 0.962·27-s + 1.67·29-s + 0.718·31-s + 0.348·33-s − 0.986·37-s + 0.312·41-s + 1.37·43-s − 0.583·47-s − 49-s + 0.280·51-s + 1.78·53-s − 1.05·57-s − 0.781·59-s − 0.640·61-s − 0.733·67-s − 0.120·69-s − 0.237·71-s − 0.936·73-s + 1.91·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.05409225922690, −14.75691127944356, −14.18122107745625, −13.81118453621213, −13.28131721471177, −12.54963863471861, −12.08898805130670, −11.71551094031351, −10.85126807298030, −10.53383294683497, −9.923837414760238, −9.133210089359412, −8.859729434933240, −8.197123632771322, −7.916951211384558, −7.013610796205740, −6.402653472653672, −6.043319423169361, −5.236338726589883, −4.468746281131751, −4.000535835089305, −3.211316752456469, −2.627185539426889, −1.980363821458357, −1.063547641156359, 0,
1.063547641156359, 1.980363821458357, 2.627185539426889, 3.211316752456469, 4.000535835089305, 4.468746281131751, 5.236338726589883, 6.043319423169361, 6.402653472653672, 7.013610796205740, 7.916951211384558, 8.197123632771322, 8.859729434933240, 9.133210089359412, 9.923837414760238, 10.53383294683497, 10.85126807298030, 11.71551094031351, 12.08898805130670, 12.54963863471861, 13.28131721471177, 13.81118453621213, 14.18122107745625, 14.75691127944356, 15.05409225922690
