| L(s) = 1 | + 3-s + 5-s − 3.37·7-s + 9-s + 0.627·11-s + 4.74·13-s + 15-s + 17-s − 0.627·19-s − 3.37·21-s + 25-s + 27-s + 2.62·29-s + 0.627·33-s − 3.37·35-s + 2.62·37-s + 4.74·39-s − 1.37·41-s − 4·43-s + 45-s + 11.3·47-s + 4.37·49-s + 51-s − 4.11·53-s + 0.627·55-s − 0.627·57-s + 2.74·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.27·7-s + 0.333·9-s + 0.189·11-s + 1.31·13-s + 0.258·15-s + 0.242·17-s − 0.144·19-s − 0.735·21-s + 0.200·25-s + 0.192·27-s + 0.487·29-s + 0.109·33-s − 0.570·35-s + 0.431·37-s + 0.759·39-s − 0.214·41-s − 0.609·43-s + 0.149·45-s + 1.65·47-s + 0.624·49-s + 0.140·51-s − 0.565·53-s + 0.0846·55-s − 0.0831·57-s + 0.357·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.434163526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434163526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 19 | \( 1 + 0.627T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 + 0.744T + 61T^{2} \) |
| 67 | \( 1 + 2.74T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5.37T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.629294784759904751423681599103, −7.74205612318810657670186338013, −6.85093448846986111948053621903, −6.27267211867829412130172921043, −5.68934042195116440685465664256, −4.51798696291463088924994511595, −3.60624628769672871015199610679, −3.08923817997205572921977951897, −2.05688727313117556416488292186, −0.882719482990355179571335541345,
0.882719482990355179571335541345, 2.05688727313117556416488292186, 3.08923817997205572921977951897, 3.60624628769672871015199610679, 4.51798696291463088924994511595, 5.68934042195116440685465664256, 6.27267211867829412130172921043, 6.85093448846986111948053621903, 7.74205612318810657670186338013, 8.629294784759904751423681599103
