| L(s) = 1 | + 4·3-s − 16·7-s + 7·9-s − 24·13-s + 12·19-s − 64·21-s − 8·27-s + 68·31-s − 88·37-s − 96·39-s + 56·43-s + 94·49-s + 48·57-s + 148·61-s − 112·63-s + 184·67-s − 112·73-s + 156·79-s − 95·81-s + 384·91-s + 272·93-s + 64·97-s + 208·103-s − 148·109-s − 352·111-s − 168·117-s + 162·121-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 2.28·7-s + 7/9·9-s − 1.84·13-s + 0.631·19-s − 3.04·21-s − 0.296·27-s + 2.19·31-s − 2.37·37-s − 2.46·39-s + 1.30·43-s + 1.91·49-s + 0.842·57-s + 2.42·61-s − 1.77·63-s + 2.74·67-s − 1.53·73-s + 1.97·79-s − 1.17·81-s + 4.21·91-s + 2.92·93-s + 0.659·97-s + 2.01·103-s − 1.35·109-s − 3.17·111-s − 1.43·117-s + 1.33·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.718426878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718426878\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 4 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 162 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 402 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1038 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 962 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3042 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4398 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3998 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2718 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 92 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7202 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3198 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15522 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.99558214838770257438958303892, −11.40335385784830607465475228971, −10.48923469525591158892951829158, −10.00816980617928662807735869112, −9.919435320945205768970169907586, −9.325249105445537376999411629286, −9.215912645048947635053377477771, −8.306418873260894727391541496925, −8.156892447225875542044634213066, −7.16441387574785811788768865694, −7.10634137949585791191249430813, −6.55938648755578187452958607733, −5.87456657182670705147906524786, −5.18529702695956408999013598437, −4.52760557085285635632207330829, −3.52706854472532155384578207998, −3.41167699065757616512106514696, −2.60169216266832697130234589791, −2.26416637353483593926308681575, −0.56330660833230031658104037037,
0.56330660833230031658104037037, 2.26416637353483593926308681575, 2.60169216266832697130234589791, 3.41167699065757616512106514696, 3.52706854472532155384578207998, 4.52760557085285635632207330829, 5.18529702695956408999013598437, 5.87456657182670705147906524786, 6.55938648755578187452958607733, 7.10634137949585791191249430813, 7.16441387574785811788768865694, 8.156892447225875542044634213066, 8.306418873260894727391541496925, 9.215912645048947635053377477771, 9.325249105445537376999411629286, 9.919435320945205768970169907586, 10.00816980617928662807735869112, 10.48923469525591158892951829158, 11.40335385784830607465475228971, 11.99558214838770257438958303892
