L(s) = 1 | − 1.61·2-s + 0.618·4-s − 2.85·7-s + 2.23·8-s − 11-s + 6.23·13-s + 4.61·14-s − 4.85·16-s + 0.618·17-s − 6.70·19-s + 1.61·22-s + 4.09·23-s − 10.0·26-s − 1.76·28-s + 1.38·29-s − 3·31-s + 3.38·32-s − 1.00·34-s + 10.2·37-s + 10.8·38-s + 3·41-s − 6·43-s − 0.618·44-s − 6.61·46-s − 11.9·47-s + 1.14·49-s + 3.85·52-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 0.309·4-s − 1.07·7-s + 0.790·8-s − 0.301·11-s + 1.72·13-s + 1.23·14-s − 1.21·16-s + 0.149·17-s − 1.53·19-s + 0.344·22-s + 0.852·23-s − 1.97·26-s − 0.333·28-s + 0.256·29-s − 0.538·31-s + 0.597·32-s − 0.171·34-s + 1.68·37-s + 1.76·38-s + 0.468·41-s − 0.914·43-s − 0.0931·44-s − 0.975·46-s − 1.74·47-s + 0.163·49-s + 0.534·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6802251722\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6802251722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 9.32T + 53T^{2} \) |
| 59 | \( 1 + 0.527T + 59T^{2} \) |
| 61 | \( 1 - 0.0901T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 5.85T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 6.90T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−8.941659373707782851068029587110, −8.323491105795828004422478385486, −7.68866108884944240775195028373, −6.45817892164130181003547533687, −6.35320534499018699352081772823, −4.97427209371664079860068142091, −3.98800753877066793284880343359, −3.11298100691947915189548158533, −1.81765316420836059210672361330, −0.62962301905626252087556351454,
0.62962301905626252087556351454, 1.81765316420836059210672361330, 3.11298100691947915189548158533, 3.98800753877066793284880343359, 4.97427209371664079860068142091, 6.35320534499018699352081772823, 6.45817892164130181003547533687, 7.68866108884944240775195028373, 8.323491105795828004422478385486, 8.941659373707782851068029587110