| L(s) = 1 | + 2.31·3-s + 1.31·5-s + 2.37·9-s − 2.07·11-s + 4.38·13-s + 3.05·15-s − 17-s + 5.45·19-s − 7.52·23-s − 3.25·25-s − 1.44·27-s + 9.33·29-s + 6.25·31-s − 4.82·33-s + 5.76·37-s + 10.1·39-s + 7.09·41-s + 8.70·43-s + 3.13·45-s + 1.93·47-s − 2.31·51-s − 8.07·53-s − 2.74·55-s + 12.6·57-s + 0.618·59-s + 8.08·61-s + 5.78·65-s + ⋯ |
| L(s) = 1 | + 1.33·3-s + 0.589·5-s + 0.792·9-s − 0.626·11-s + 1.21·13-s + 0.789·15-s − 0.242·17-s + 1.25·19-s − 1.56·23-s − 0.651·25-s − 0.277·27-s + 1.73·29-s + 1.12·31-s − 0.839·33-s + 0.946·37-s + 1.62·39-s + 1.10·41-s + 1.32·43-s + 0.467·45-s + 0.282·47-s − 0.324·51-s − 1.10·53-s − 0.369·55-s + 1.67·57-s + 0.0805·59-s + 1.03·61-s + 0.716·65-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
| \(L(1)\) |
\(\approx\) |
\(3.575692299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575692299\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.31T + 3T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 29 | \( 1 - 9.33T + 29T^{2} \) |
| 31 | \( 1 - 6.25T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 - 1.93T + 47T^{2} \) |
| 53 | \( 1 + 8.07T + 53T^{2} \) |
| 59 | \( 1 - 0.618T + 59T^{2} \) |
| 61 | \( 1 - 8.08T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 - 1.21T + 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−8.520226618363990100737577301546, −8.024660951844058239075576698751, −7.45608573786980079464426188961, −6.20734441029112101998305528782, −5.83840455631911393103599597063, −4.59531615731497009614130510567, −3.79596337858987000807788266031, −2.87913394486278734258340316185, −2.28673583908713919429302980258, −1.13320366624766713639877540761,
1.13320366624766713639877540761, 2.28673583908713919429302980258, 2.87913394486278734258340316185, 3.79596337858987000807788266031, 4.59531615731497009614130510567, 5.83840455631911393103599597063, 6.20734441029112101998305528782, 7.45608573786980079464426188961, 8.024660951844058239075576698751, 8.520226618363990100737577301546
