| L(s) = 1 | − 2.43·3-s − 3.60·7-s + 2.90·9-s − 2.37·11-s + 3.96·13-s − 4.28·17-s + 19-s + 8.76·21-s + 1.07·23-s + 0.227·27-s + 9.47·29-s + 6.38·31-s + 5.78·33-s − 2.04·37-s − 9.63·39-s − 4.38·41-s − 7.86·43-s − 3.83·47-s + 6.01·49-s + 10.4·51-s + 11.7·53-s − 2.43·57-s + 4.59·59-s + 6.62·61-s − 10.4·63-s − 7.02·67-s − 2.60·69-s + ⋯ |
| L(s) = 1 | − 1.40·3-s − 1.36·7-s + 0.968·9-s − 0.717·11-s + 1.10·13-s − 1.03·17-s + 0.229·19-s + 1.91·21-s + 0.223·23-s + 0.0437·27-s + 1.75·29-s + 1.14·31-s + 1.00·33-s − 0.336·37-s − 1.54·39-s − 0.684·41-s − 1.19·43-s − 0.559·47-s + 0.859·49-s + 1.45·51-s + 1.61·53-s − 0.321·57-s + 0.598·59-s + 0.848·61-s − 1.32·63-s − 0.858·67-s − 0.314·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.43T + 3T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 - 3.96T + 13T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 - 9.47T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 + 4.38T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 4.59T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 - 4.99T + 71T^{2} \) |
| 73 | \( 1 - 2.93T + 73T^{2} \) |
| 79 | \( 1 - 0.860T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−8.269804930161372473796696041868, −6.87214967880358875786595777107, −6.65194225880039696802627093771, −5.99990796320943991407626417545, −5.23459269534444572600849741679, −4.48776180496958968650406602687, −3.46172131490042659473916346950, −2.58811194199752644047495856550, −1.02054784796983291449167355497, 0,
1.02054784796983291449167355497, 2.58811194199752644047495856550, 3.46172131490042659473916346950, 4.48776180496958968650406602687, 5.23459269534444572600849741679, 5.99990796320943991407626417545, 6.65194225880039696802627093771, 6.87214967880358875786595777107, 8.269804930161372473796696041868
