L(s) = 1 | + 2.27·3-s + 2.37·5-s − 1.75·7-s + 2.16·9-s − 6.02·11-s + 5.29·13-s + 5.40·15-s + 7.04·17-s + 6.15·19-s − 3.99·21-s − 5.05·23-s + 0.659·25-s − 1.89·27-s + 2.49·29-s − 4.57·31-s − 13.6·33-s − 4.17·35-s + 8.89·37-s + 12.0·39-s + 10.6·41-s − 11.1·43-s + 5.15·45-s + 10.2·47-s − 3.91·49-s + 16.0·51-s − 2.73·53-s − 14.3·55-s + ⋯ |
L(s) = 1 | + 1.31·3-s + 1.06·5-s − 0.663·7-s + 0.721·9-s − 1.81·11-s + 1.46·13-s + 1.39·15-s + 1.70·17-s + 1.41·19-s − 0.870·21-s − 1.05·23-s + 0.131·25-s − 0.365·27-s + 0.463·29-s − 0.822·31-s − 2.38·33-s − 0.706·35-s + 1.46·37-s + 1.92·39-s + 1.65·41-s − 1.70·43-s + 0.767·45-s + 1.49·47-s − 0.559·49-s + 2.24·51-s − 0.375·53-s − 1.93·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.031860999\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.031860999\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 11 | \( 1 + 6.02T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 - 7.04T + 17T^{2} \) |
| 19 | \( 1 - 6.15T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 2.73T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 8.80T + 61T^{2} \) |
| 67 | \( 1 - 9.78T + 67T^{2} \) |
| 71 | \( 1 - 8.39T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 - 9.64T + 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−8.004260781007478711426630494490, −7.43403086072418446770826985310, −6.34249035582524753945323652060, −5.60049622948614623748097550151, −5.39539533194850406967761227891, −3.94502262333074983283993694204, −3.29014497847122268008400560513, −2.75483228907658030106798440769, −2.01415892705559589258573495937, −0.949636993247834525006839009687,
0.949636993247834525006839009687, 2.01415892705559589258573495937, 2.75483228907658030106798440769, 3.29014497847122268008400560513, 3.94502262333074983283993694204, 5.39539533194850406967761227891, 5.60049622948614623748097550151, 6.34249035582524753945323652060, 7.43403086072418446770826985310, 8.004260781007478711426630494490