| L(s) = 1 | − 2.80·2-s − 0.314·3-s + 5.88·4-s − 0.925·5-s + 0.882·6-s + 2.15·7-s − 10.9·8-s − 2.90·9-s + 2.59·10-s + 11-s − 1.84·12-s + 1.28·13-s − 6.03·14-s + 0.290·15-s + 18.8·16-s + 3.37·17-s + 8.14·18-s + 2.49·19-s − 5.44·20-s − 0.675·21-s − 2.80·22-s − 4.09·23-s + 3.42·24-s − 4.14·25-s − 3.59·26-s + 1.85·27-s + 12.6·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s − 0.181·3-s + 2.94·4-s − 0.413·5-s + 0.360·6-s + 0.812·7-s − 3.85·8-s − 0.967·9-s + 0.821·10-s + 0.301·11-s − 0.533·12-s + 0.355·13-s − 1.61·14-s + 0.0751·15-s + 4.71·16-s + 0.819·17-s + 1.92·18-s + 0.572·19-s − 1.21·20-s − 0.147·21-s − 0.598·22-s − 0.853·23-s + 0.699·24-s − 0.828·25-s − 0.705·26-s + 0.356·27-s + 2.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5453703681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5453703681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 3 | \( 1 + 0.314T + 3T^{2} \) |
| 5 | \( 1 + 0.925T + 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 3.04T + 29T^{2} \) |
| 31 | \( 1 - 4.27T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 8.77T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 + 1.94T + 47T^{2} \) |
| 53 | \( 1 - 3.62T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 67 | \( 1 - 7.30T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 - 9.85T + 73T^{2} \) |
| 79 | \( 1 - 0.979T + 79T^{2} \) |
| 83 | \( 1 + 1.70T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−10.39690768163884941373530650319, −9.591109968682619685168660703598, −8.628429648834243065928666746048, −8.112029988776276119231355395771, −7.44259819049664991904377500673, −6.33093330763506556818512766751, −5.48767972979002246041862624965, −3.50107726919693237420040547751, −2.20498099301108499224396318873, −0.853061118508044469175474566975,
0.853061118508044469175474566975, 2.20498099301108499224396318873, 3.50107726919693237420040547751, 5.48767972979002246041862624965, 6.33093330763506556818512766751, 7.44259819049664991904377500673, 8.112029988776276119231355395771, 8.628429648834243065928666746048, 9.591109968682619685168660703598, 10.39690768163884941373530650319
