L(s) = 1 | + 1.46·2-s + 0.154·4-s − 3.84·5-s + 2.46·7-s − 2.70·8-s − 5.64·10-s − 11-s − 4.64·13-s + 3.61·14-s − 4.28·16-s − 5.70·17-s + 6.71·19-s − 0.594·20-s − 1.46·22-s − 2.93·23-s + 9.76·25-s − 6.82·26-s + 0.381·28-s − 5.67·29-s + 0.232·31-s − 0.874·32-s − 8.37·34-s − 9.46·35-s + 5.55·37-s + 9.85·38-s + 10.4·40-s + 4.35·41-s + ⋯ |
L(s) = 1 | + 1.03·2-s + 0.0774·4-s − 1.71·5-s + 0.930·7-s − 0.957·8-s − 1.78·10-s − 0.301·11-s − 1.28·13-s + 0.965·14-s − 1.07·16-s − 1.38·17-s + 1.53·19-s − 0.133·20-s − 0.312·22-s − 0.612·23-s + 1.95·25-s − 1.33·26-s + 0.0720·28-s − 1.05·29-s + 0.0417·31-s − 0.154·32-s − 1.43·34-s − 1.59·35-s + 0.912·37-s + 1.59·38-s + 1.64·40-s + 0.680·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.266085115\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.266085115\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 5 | \( 1 + 3.84T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 - 0.232T + 31T^{2} \) |
| 37 | \( 1 - 5.55T + 37T^{2} \) |
| 41 | \( 1 - 4.35T + 41T^{2} \) |
| 43 | \( 1 + 9.35T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 0.299T + 71T^{2} \) |
| 73 | \( 1 + 2.65T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 7.42T + 83T^{2} \) |
| 89 | \( 1 - 3.40T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.022905784675026991812276360126, −7.32736928586743365135050235270, −6.79644190145913840437622181079, −5.57143178146289976955954786081, −4.95574733148785872949731921913, −4.48565674430134741828393592194, −3.85030570135814113276858260392, −3.08819759185783626126966260949, −2.16149840244085666868950148157, −0.48212482915960218693547648457,
0.48212482915960218693547648457, 2.16149840244085666868950148157, 3.08819759185783626126966260949, 3.85030570135814113276858260392, 4.48565674430134741828393592194, 4.95574733148785872949731921913, 5.57143178146289976955954786081, 6.79644190145913840437622181079, 7.32736928586743365135050235270, 8.022905784675026991812276360126