| L(s) = 1 | − 0.777·2-s − 1.39·4-s − 0.966·5-s − 3.17·7-s + 2.64·8-s + 0.751·10-s + 6.34·11-s − 5.47·13-s + 2.47·14-s + 0.734·16-s − 3.46·17-s − 1.75·19-s + 1.34·20-s − 4.93·22-s − 0.492·23-s − 4.06·25-s + 4.26·26-s + 4.43·28-s − 3.20·29-s + 9.29·31-s − 5.85·32-s + 2.69·34-s + 3.07·35-s − 6.60·37-s + 1.36·38-s − 2.55·40-s − 3.28·41-s + ⋯ |
| L(s) = 1 | − 0.550·2-s − 0.697·4-s − 0.432·5-s − 1.20·7-s + 0.933·8-s + 0.237·10-s + 1.91·11-s − 1.51·13-s + 0.660·14-s + 0.183·16-s − 0.841·17-s − 0.401·19-s + 0.301·20-s − 1.05·22-s − 0.102·23-s − 0.813·25-s + 0.836·26-s + 0.837·28-s − 0.594·29-s + 1.66·31-s − 1.03·32-s + 0.462·34-s + 0.519·35-s − 1.08·37-s + 0.221·38-s − 0.403·40-s − 0.512·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5753464635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5753464635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 0.777T + 2T^{2} \) |
| 5 | \( 1 + 0.966T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 1.75T + 19T^{2} \) |
| 23 | \( 1 + 0.492T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 - 9.29T + 31T^{2} \) |
| 37 | \( 1 + 6.60T + 37T^{2} \) |
| 41 | \( 1 + 3.28T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 6.59T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 8.13T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 6.88T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−9.182165791740027190877922348523, −8.521862065092882071789812177942, −7.54583415932770890673489926700, −6.81406321752615875095115302174, −6.18321858810582323608400296366, −4.87375001090152749267364024912, −4.14377946930510622606351283948, −3.44373224181989468276095311920, −2.03585999885997693801903982276, −0.53097080855103736771772075999,
0.53097080855103736771772075999, 2.03585999885997693801903982276, 3.44373224181989468276095311920, 4.14377946930510622606351283948, 4.87375001090152749267364024912, 6.18321858810582323608400296366, 6.81406321752615875095115302174, 7.54583415932770890673489926700, 8.521862065092882071789812177942, 9.182165791740027190877922348523
