| L(s) = 1 | + 3.06·3-s + 5-s + 3.30·7-s + 6.36·9-s + 0.653·11-s − 0.734·13-s + 3.06·15-s − 0.0784·17-s − 4.58·19-s + 10.0·21-s + 3.06·23-s + 25-s + 10.2·27-s − 29-s − 1.51·31-s + 1.99·33-s + 3.30·35-s + 7.28·37-s − 2.24·39-s − 12.0·41-s + 7.71·43-s + 6.36·45-s + 4.98·47-s + 3.89·49-s − 0.240·51-s − 2.05·53-s + 0.653·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s + 0.447·5-s + 1.24·7-s + 2.12·9-s + 0.197·11-s − 0.203·13-s + 0.790·15-s − 0.0190·17-s − 1.05·19-s + 2.20·21-s + 0.638·23-s + 0.200·25-s + 1.98·27-s − 0.185·29-s − 0.271·31-s + 0.348·33-s + 0.557·35-s + 1.19·37-s − 0.360·39-s − 1.88·41-s + 1.17·43-s + 0.948·45-s + 0.727·47-s + 0.555·49-s − 0.0336·51-s − 0.281·53-s + 0.0881·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.016292502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.016292502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 3.06T + 3T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 0.653T + 11T^{2} \) |
| 13 | \( 1 + 0.734T + 13T^{2} \) |
| 17 | \( 1 + 0.0784T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 - 7.28T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 - 2.00T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 + 4.19T + 67T^{2} \) |
| 71 | \( 1 + 4.41T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 + 2.57T + 79T^{2} \) |
| 83 | \( 1 - 5.54T + 83T^{2} \) |
| 89 | \( 1 + 0.844T + 89T^{2} \) |
| 97 | \( 1 + 7.33T + 97T^{2} \) |
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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
−8.421397577006797143466389410158, −7.73824312011636357409669394011, −7.17500963238219135344999309915, −6.26606819757362781743852781465, −5.14423620097040395266211967029, −4.43700456039914791336612828620, −3.72672698713726199518013356545, −2.70054138675233019431048599546, −2.07987367981379707440311871402, −1.30807927057712731587098781030,
1.30807927057712731587098781030, 2.07987367981379707440311871402, 2.70054138675233019431048599546, 3.72672698713726199518013356545, 4.43700456039914791336612828620, 5.14423620097040395266211967029, 6.26606819757362781743852781465, 7.17500963238219135344999309915, 7.73824312011636357409669394011, 8.421397577006797143466389410158
