| L(s) = 1 | + 3-s − 4.13·7-s + 9-s − 3.76·11-s − 0.698·13-s − 5.29·17-s + 5.73·19-s − 4.13·21-s + 5.46·23-s + 27-s − 7.02·29-s − 10.0·31-s − 3.76·33-s + 5.78·37-s − 0.698·39-s − 6.59·41-s + 4.79·43-s + 9.67·47-s + 10.0·49-s − 5.29·51-s − 3.39·53-s + 5.73·57-s + 0.745·59-s + 11.8·61-s − 4.13·63-s − 4.79·67-s + 5.46·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.56·7-s + 0.333·9-s − 1.13·11-s − 0.193·13-s − 1.28·17-s + 1.31·19-s − 0.901·21-s + 1.13·23-s + 0.192·27-s − 1.30·29-s − 1.81·31-s − 0.655·33-s + 0.951·37-s − 0.111·39-s − 1.03·41-s + 0.731·43-s + 1.41·47-s + 1.43·49-s − 0.741·51-s − 0.466·53-s + 0.759·57-s + 0.0970·59-s + 1.51·61-s − 0.520·63-s − 0.586·67-s + 0.658·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.326518471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.326518471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 0.698T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 5.78T + 37T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 - 4.79T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 - 0.745T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 + 3.14T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 - 8.44T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.70291819689439517148487122078, −7.24188341089739858723535608356, −6.69769253004126368519979087497, −5.70961046375705671256880845234, −5.21088928231245928921099473642, −4.13845888619265971334454210198, −3.37783132587487620235586669647, −2.80961934175106117699056870285, −2.03902433819952127039708653118, −0.52846444007236409334725216849,
0.52846444007236409334725216849, 2.03902433819952127039708653118, 2.80961934175106117699056870285, 3.37783132587487620235586669647, 4.13845888619265971334454210198, 5.21088928231245928921099473642, 5.70961046375705671256880845234, 6.69769253004126368519979087497, 7.24188341089739858723535608356, 7.70291819689439517148487122078
