| L(s) = 1 | − 5-s + 3.83·7-s + 1.14·11-s − 3.53·13-s + 6.97·17-s − 19-s + 8.97·23-s + 25-s − 0.853·29-s − 4.39·31-s − 3.83·35-s + 1.83·37-s + 8.51·41-s − 10.1·43-s − 0.978·47-s + 7.68·49-s − 6.97·53-s − 1.14·55-s + 6.29·59-s + 10.0·61-s + 3.53·65-s + 0.585·67-s + 11.0·71-s + 7.37·73-s + 4.39·77-s − 12.6·83-s − 6.97·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.44·7-s + 0.345·11-s − 0.981·13-s + 1.69·17-s − 0.229·19-s + 1.87·23-s + 0.200·25-s − 0.158·29-s − 0.789·31-s − 0.647·35-s + 0.301·37-s + 1.33·41-s − 1.54·43-s − 0.142·47-s + 1.09·49-s − 0.958·53-s − 0.154·55-s + 0.819·59-s + 1.28·61-s + 0.439·65-s + 0.0715·67-s + 1.31·71-s + 0.862·73-s + 0.500·77-s − 1.38·83-s − 0.756·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.426185825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.426185825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 23 | \( 1 - 8.97T + 23T^{2} \) |
| 29 | \( 1 + 0.853T + 29T^{2} \) |
| 31 | \( 1 + 4.39T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 0.978T + 47T^{2} \) |
| 53 | \( 1 + 6.97T + 53T^{2} \) |
| 59 | \( 1 - 6.29T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 0.585T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 7.37T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 7.93T + 89T^{2} \) |
| 97 | \( 1 - 1.24T + 97T^{2} \) |
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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
−8.027557046110247770422400845321, −7.30039242726101653940181521356, −6.83882305283753704274512218348, −5.56335139314094948541107678899, −5.14904413436863669482918480116, −4.48801080342808615949705689509, −3.60757074962314543609952932569, −2.75632734145365127872300602889, −1.69760178181272573333476108138, −0.849480339102826374474228896215,
0.849480339102826374474228896215, 1.69760178181272573333476108138, 2.75632734145365127872300602889, 3.60757074962314543609952932569, 4.48801080342808615949705689509, 5.14904413436863669482918480116, 5.56335139314094948541107678899, 6.83882305283753704274512218348, 7.30039242726101653940181521356, 8.027557046110247770422400845321
