L(s) = 1 | − 3.67·5-s + 4.79·7-s − 1.91·11-s + 3.05·13-s + 2.92·17-s − 8.17·19-s − 6.97·23-s + 8.49·25-s + 3.87·29-s + 0.104·31-s − 17.6·35-s − 1.37·37-s + 10.4·41-s − 5.08·43-s − 10.8·47-s + 16.0·49-s − 7.42·53-s + 7.03·55-s − 0.535·59-s + 8.80·61-s − 11.2·65-s − 8.50·67-s + 5.00·71-s + 8.80·73-s − 9.18·77-s − 6.76·79-s − 10.3·83-s + ⋯ |
L(s) = 1 | − 1.64·5-s + 1.81·7-s − 0.577·11-s + 0.847·13-s + 0.710·17-s − 1.87·19-s − 1.45·23-s + 1.69·25-s + 0.720·29-s + 0.0187·31-s − 2.97·35-s − 0.226·37-s + 1.63·41-s − 0.774·43-s − 1.57·47-s + 2.28·49-s − 1.01·53-s + 0.948·55-s − 0.0696·59-s + 1.12·61-s − 1.39·65-s − 1.03·67-s + 0.594·71-s + 1.03·73-s − 1.04·77-s − 0.761·79-s − 1.13·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 - 4.79T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 19 | \( 1 + 8.17T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 - 0.104T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 7.42T + 53T^{2} \) |
| 59 | \( 1 + 0.535T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 + 8.50T + 67T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 - 8.80T + 73T^{2} \) |
| 79 | \( 1 + 6.76T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.978593307315781983432266928738, −7.27838099887826477650939161377, −6.33616122184797873324052102264, −5.46864825332516911413558178206, −4.49771587407210907850533417890, −4.29176813107568924899521969117, −3.41342618228403518022843689483, −2.25262196055315800254741644641, −1.28322454468950760554651346196, 0,
1.28322454468950760554651346196, 2.25262196055315800254741644641, 3.41342618228403518022843689483, 4.29176813107568924899521969117, 4.49771587407210907850533417890, 5.46864825332516911413558178206, 6.33616122184797873324052102264, 7.27838099887826477650939161377, 7.978593307315781983432266928738