L(s) = 1 | + 8.27·5-s − 8.82·7-s + 9·9-s + 17.3·11-s − 33.9·17-s − 19·19-s − 30·23-s + 43.4·25-s − 73.0·35-s + 31.1·43-s + 74.4·45-s + 11.5·47-s + 28.8·49-s + 143.·55-s − 108.·61-s − 79.4·63-s + 137.·73-s − 153.·77-s + 81·81-s + 90·83-s − 280.·85-s − 157.·95-s + 156.·99-s − 102·101-s − 248.·115-s + 299.·119-s + ⋯ |
L(s) = 1 | + 1.65·5-s − 1.26·7-s + 9-s + 1.57·11-s − 1.99·17-s − 19-s − 1.30·23-s + 1.73·25-s − 2.08·35-s + 0.725·43-s + 1.65·45-s + 0.246·47-s + 0.589·49-s + 2.61·55-s − 1.77·61-s − 1.26·63-s + 1.87·73-s − 1.99·77-s + 81-s + 1.08·83-s − 3.30·85-s − 1.65·95-s + 1.57·99-s − 1.00·101-s − 2.15·115-s + 2.51·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.476047574\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.476047574\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
good | 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 - 8.27T + 25T^{2} \) |
| 7 | \( 1 + 8.82T + 49T^{2} \) |
| 11 | \( 1 - 17.3T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 33.9T + 289T^{2} \) |
| 23 | \( 1 + 30T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 31.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 11.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 90T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−13.98954748611141130826616464395, −13.25793806686218187541230197635, −12.40791984745472605653512357752, −10.66384767785405434824367159722, −9.623893377756596418000609728203, −9.062644504887732010909784046997, −6.63433232389462037367473591257, −6.27859806420955956445996812767, −4.19099285709592402724125876299, −2.01462065768240650349644573681,
2.01462065768240650349644573681, 4.19099285709592402724125876299, 6.27859806420955956445996812767, 6.63433232389462037367473591257, 9.062644504887732010909784046997, 9.623893377756596418000609728203, 10.66384767785405434824367159722, 12.40791984745472605653512357752, 13.25793806686218187541230197635, 13.98954748611141130826616464395