L(s) = 1 | + 0.530·2-s − 1.71·4-s − 3.35·5-s + 3.91·7-s − 1.97·8-s − 1.78·10-s + 0.0181·11-s − 5.92·13-s + 2.07·14-s + 2.39·16-s − 2.80·17-s + 8.42·19-s + 5.77·20-s + 0.00961·22-s + 7.23·23-s + 6.27·25-s − 3.14·26-s − 6.72·28-s − 0.873·29-s + 0.822·31-s + 5.21·32-s − 1.48·34-s − 13.1·35-s − 0.327·37-s + 4.47·38-s + 6.62·40-s − 8.66·41-s + ⋯ |
L(s) = 1 | + 0.375·2-s − 0.859·4-s − 1.50·5-s + 1.47·7-s − 0.697·8-s − 0.563·10-s + 0.00546·11-s − 1.64·13-s + 0.555·14-s + 0.597·16-s − 0.679·17-s + 1.93·19-s + 1.29·20-s + 0.00205·22-s + 1.50·23-s + 1.25·25-s − 0.616·26-s − 1.27·28-s − 0.162·29-s + 0.147·31-s + 0.921·32-s − 0.254·34-s − 2.22·35-s − 0.0538·37-s + 0.725·38-s + 1.04·40-s − 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 2 | \( 1 - 0.530T + 2T^{2} \) |
| 5 | \( 1 + 3.35T + 5T^{2} \) |
| 7 | \( 1 - 3.91T + 7T^{2} \) |
| 11 | \( 1 - 0.0181T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 19 | \( 1 - 8.42T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 + 0.873T + 29T^{2} \) |
| 31 | \( 1 - 0.822T + 31T^{2} \) |
| 37 | \( 1 + 0.327T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 + 1.00T + 43T^{2} \) |
| 47 | \( 1 + 7.71T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 4.56T + 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 5.83T + 73T^{2} \) |
| 79 | \( 1 + 0.209T + 79T^{2} \) |
| 83 | \( 1 + 0.907T + 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 + 6.76T + 97T^{2} \) |
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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
−8.065545875080580670036950950400, −7.44628718536472245720479571284, −6.91489329063998436344138069345, −5.21586915498589225304870276214, −5.07326239713300253600535369295, −4.44184303725600205780088386766, −3.57465715551004325325797436639, −2.75125861537899314184217098913, −1.21730340557142594813163462689, 0,
1.21730340557142594813163462689, 2.75125861537899314184217098913, 3.57465715551004325325797436639, 4.44184303725600205780088386766, 5.07326239713300253600535369295, 5.21586915498589225304870276214, 6.91489329063998436344138069345, 7.44628718536472245720479571284, 8.065545875080580670036950950400