L(s) = 1 | − 2.41·2-s − 2.16·4-s − 35.7·7-s + 24.5·8-s − 11·11-s + 41.3·13-s + 86.4·14-s − 41.9·16-s + 13.1·17-s − 30.3·19-s + 26.5·22-s − 18.0·23-s − 99.7·26-s + 77.6·28-s − 106.·29-s + 125.·31-s − 95.2·32-s − 31.8·34-s − 71.8·37-s + 73.1·38-s + 203.·41-s − 283.·43-s + 23.8·44-s + 43.5·46-s − 60.0·47-s + 938.·49-s − 89.6·52-s + ⋯ |
L(s) = 1 | − 0.853·2-s − 0.271·4-s − 1.93·7-s + 1.08·8-s − 0.301·11-s + 0.881·13-s + 1.65·14-s − 0.655·16-s + 0.188·17-s − 0.365·19-s + 0.257·22-s − 0.163·23-s − 0.752·26-s + 0.524·28-s − 0.683·29-s + 0.727·31-s − 0.525·32-s − 0.160·34-s − 0.319·37-s + 0.312·38-s + 0.775·41-s − 1.00·43-s + 0.0817·44-s + 0.139·46-s − 0.186·47-s + 2.73·49-s − 0.239·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3659664015\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3659664015\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 + 2.41T + 8T^{2} \) |
| 7 | \( 1 + 35.7T + 343T^{2} \) |
| 13 | \( 1 - 41.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 13.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 18.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 71.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 203.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 283.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 60.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 352.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 9.68T + 2.05e5T^{2} \) |
| 61 | \( 1 + 200.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 411.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 790.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 666.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 496.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.40e3T + 9.12e5T^{2} \) |
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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
−8.819887367134401613353421773952, −7.938775656856018597322986973773, −7.14557439015728789658310988217, −6.34622949879230183249123334036, −5.70425260514419102828067672247, −4.47885043441857919289595998516, −3.63497231211987200700872471291, −2.83719727514061050367060988467, −1.46772490343500250360342531015, −0.31693162375796935246456005950,
0.31693162375796935246456005950, 1.46772490343500250360342531015, 2.83719727514061050367060988467, 3.63497231211987200700872471291, 4.47885043441857919289595998516, 5.70425260514419102828067672247, 6.34622949879230183249123334036, 7.14557439015728789658310988217, 7.938775656856018597322986973773, 8.819887367134401613353421773952