| L(s) = 1 | + 3.11·3-s − 0.181·5-s − 2.43·7-s + 6.69·9-s + 1.74·11-s − 4.15·13-s − 0.564·15-s + 4.74·17-s + 4.56·19-s − 7.59·21-s − 0.0747·23-s − 4.96·25-s + 11.5·27-s + 5.83·29-s − 2.51·31-s + 5.44·33-s + 0.442·35-s + 3.72·37-s − 12.9·39-s + 2.08·41-s − 0.814·43-s − 1.21·45-s − 1.82·47-s − 1.05·49-s + 14.7·51-s + 13.9·53-s − 0.317·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 0.0810·5-s − 0.921·7-s + 2.23·9-s + 0.527·11-s − 1.15·13-s − 0.145·15-s + 1.15·17-s + 1.04·19-s − 1.65·21-s − 0.0155·23-s − 0.993·25-s + 2.21·27-s + 1.08·29-s − 0.451·31-s + 0.948·33-s + 0.0747·35-s + 0.613·37-s − 2.07·39-s + 0.326·41-s − 0.124·43-s − 0.180·45-s − 0.266·47-s − 0.150·49-s + 2.07·51-s + 1.92·53-s − 0.0427·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.838210486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.838210486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 5 | \( 1 + 0.181T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4.56T + 19T^{2} \) |
| 23 | \( 1 + 0.0747T + 23T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 - 3.72T + 37T^{2} \) |
| 41 | \( 1 - 2.08T + 41T^{2} \) |
| 43 | \( 1 + 0.814T + 43T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 7.56T + 79T^{2} \) |
| 83 | \( 1 + 0.177T + 83T^{2} \) |
| 89 | \( 1 + 0.488T + 89T^{2} \) |
| 97 | \( 1 - 5.46T + 97T^{2} \) |
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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
−8.142641741769241993287472397402, −7.31466292060828220070813536896, −7.13259248802882215447568705332, −6.03513720488394863732280621909, −5.09360407549060472977475273109, −4.07267026705832441396796656403, −3.50536998782448112874407574235, −2.85409961928572713200773376970, −2.15772642323910045527057644110, −0.967096649237018675589984284156,
0.967096649237018675589984284156, 2.15772642323910045527057644110, 2.85409961928572713200773376970, 3.50536998782448112874407574235, 4.07267026705832441396796656403, 5.09360407549060472977475273109, 6.03513720488394863732280621909, 7.13259248802882215447568705332, 7.31466292060828220070813536896, 8.142641741769241993287472397402
