| L(s) = 1 | + 2.61·3-s + 3.61·5-s + 3.85·9-s + 1.23·13-s + 9.47·15-s + 17-s + 6.47·19-s + 0.763·23-s + 8.09·25-s + 2.23·27-s + 7.23·29-s − 10.5·31-s − 9.70·37-s + 3.23·39-s − 7.85·41-s − 1.85·43-s + 13.9·45-s − 6.94·47-s + 2.61·51-s − 3.32·53-s + 16.9·57-s + 0.763·59-s − 7.56·61-s + 4.47·65-s + 6.09·67-s + 2·69-s − 9.23·71-s + ⋯ |
| L(s) = 1 | + 1.51·3-s + 1.61·5-s + 1.28·9-s + 0.342·13-s + 2.44·15-s + 0.242·17-s + 1.48·19-s + 0.159·23-s + 1.61·25-s + 0.430·27-s + 1.34·29-s − 1.89·31-s − 1.59·37-s + 0.518·39-s − 1.22·41-s − 0.282·43-s + 2.07·45-s − 1.01·47-s + 0.366·51-s − 0.456·53-s + 2.24·57-s + 0.0994·59-s − 0.968·61-s + 0.554·65-s + 0.744·67-s + 0.240·69-s − 1.09·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.658453326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.658453326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 0.763T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 + 7.85T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 + 3.32T + 53T^{2} \) |
| 59 | \( 1 - 0.763T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + 9.23T + 71T^{2} \) |
| 73 | \( 1 - 3.85T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 8.76T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−8.879013673811184043608678615488, −8.016424666891161792444779543259, −7.21255838740300075668991186632, −6.47961626777573312058944839933, −5.52164488398356962041781308158, −4.91911535703869940747229027303, −3.50082886072682046288404034446, −3.07165925805224840936043531837, −2.02095606936778740314664235010, −1.43154103520227827629246674825,
1.43154103520227827629246674825, 2.02095606936778740314664235010, 3.07165925805224840936043531837, 3.50082886072682046288404034446, 4.91911535703869940747229027303, 5.52164488398356962041781308158, 6.47961626777573312058944839933, 7.21255838740300075668991186632, 8.016424666891161792444779543259, 8.879013673811184043608678615488
