| L(s) = 1 | − 1.58·2-s + 0.524·4-s + 1.06·5-s − 3.24·7-s + 2.34·8-s − 1.69·10-s − 3.21·11-s − 2.29·13-s + 5.15·14-s − 4.77·16-s − 3.37·17-s − 0.226·19-s + 0.558·20-s + 5.11·22-s − 3.86·25-s + 3.64·26-s − 1.70·28-s + 1.67·29-s − 5.68·31-s + 2.89·32-s + 5.37·34-s − 3.45·35-s + 6.77·37-s + 0.359·38-s + 2.49·40-s − 2.75·41-s + 0.337·43-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.262·4-s + 0.475·5-s − 1.22·7-s + 0.828·8-s − 0.534·10-s − 0.970·11-s − 0.636·13-s + 1.37·14-s − 1.19·16-s − 0.819·17-s − 0.0519·19-s + 0.124·20-s + 1.09·22-s − 0.773·25-s + 0.715·26-s − 0.321·28-s + 0.311·29-s − 1.02·31-s + 0.512·32-s + 0.920·34-s − 0.583·35-s + 1.11·37-s + 0.0583·38-s + 0.394·40-s − 0.429·41-s + 0.0514·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3294952961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3294952961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 + 0.226T + 19T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 + 5.68T + 31T^{2} \) |
| 37 | \( 1 - 6.77T + 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 - 0.337T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.203101756663260882945304140573, −7.85262391448445245609065721205, −6.86423431043615944111534719842, −6.42138045375427837051308670125, −5.38738107472228751831581016712, −4.68371956020111573314706011939, −3.61888651947782775647444621948, −2.60423699642721408860701859133, −1.83745018679551445332744703618, −0.35908357629509650804918401262,
0.35908357629509650804918401262, 1.83745018679551445332744703618, 2.60423699642721408860701859133, 3.61888651947782775647444621948, 4.68371956020111573314706011939, 5.38738107472228751831581016712, 6.42138045375427837051308670125, 6.86423431043615944111534719842, 7.85262391448445245609065721205, 8.203101756663260882945304140573
