L(s) = 1 | − 1.49·3-s − 0.747·5-s + 4.82·7-s − 0.767·9-s − 4.01·11-s − 0.852·13-s + 1.11·15-s + 0.486·17-s − 8.29·19-s − 7.20·21-s − 7.15·23-s − 4.44·25-s + 5.62·27-s − 10.0·29-s + 8.27·31-s + 5.99·33-s − 3.60·35-s + 2.91·37-s + 1.27·39-s + 5.00·41-s + 6.36·43-s + 0.573·45-s − 3.87·47-s + 16.2·49-s − 0.727·51-s + 8.86·53-s + 3.00·55-s + ⋯ |
L(s) = 1 | − 0.862·3-s − 0.334·5-s + 1.82·7-s − 0.255·9-s − 1.20·11-s − 0.236·13-s + 0.288·15-s + 0.118·17-s − 1.90·19-s − 1.57·21-s − 1.49·23-s − 0.888·25-s + 1.08·27-s − 1.87·29-s + 1.48·31-s + 1.04·33-s − 0.609·35-s + 0.478·37-s + 0.204·39-s + 0.780·41-s + 0.971·43-s + 0.0855·45-s − 0.565·47-s + 2.32·49-s − 0.101·51-s + 1.21·53-s + 0.404·55-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
\(L(1)\) |
\(\approx\) |
\(0.9060220816\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9060220816\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 + 0.747T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 + 0.852T + 13T^{2} \) |
| 17 | \( 1 - 0.486T + 17T^{2} \) |
| 19 | \( 1 + 8.29T + 19T^{2} \) |
| 23 | \( 1 + 7.15T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 8.27T + 31T^{2} \) |
| 37 | \( 1 - 2.91T + 37T^{2} \) |
| 41 | \( 1 - 5.00T + 41T^{2} \) |
| 43 | \( 1 - 6.36T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 - 8.86T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 6.20T + 61T^{2} \) |
| 67 | \( 1 - 8.17T + 67T^{2} \) |
| 71 | \( 1 - 9.36T + 71T^{2} \) |
| 73 | \( 1 - 7.81T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 2.87T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.086491779745258416013803396725, −7.63141499203653186794804254921, −6.54336159303202314991263849172, −5.71445785381463980545333289946, −5.34880486486320782334451784020, −4.45772138005618609472648301194, −4.04097085952590477665894978440, −2.46315821247626718489114940627, −1.94637981377466869949345060179, −0.50302354149717945573181409993,
0.50302354149717945573181409993, 1.94637981377466869949345060179, 2.46315821247626718489114940627, 4.04097085952590477665894978440, 4.45772138005618609472648301194, 5.34880486486320782334451784020, 5.71445785381463980545333289946, 6.54336159303202314991263849172, 7.63141499203653186794804254921, 8.086491779745258416013803396725