| L(s) = 1 | + 3-s + 0.951·5-s + 1.28·7-s + 9-s + 0.951·11-s − 2.82·13-s + 0.951·15-s + 1.31·17-s − 5.94·19-s + 1.28·21-s + 2.56·23-s − 4.09·25-s + 27-s − 2.48·29-s − 0.984·31-s + 0.951·33-s + 1.21·35-s − 7.81·37-s − 2.82·39-s − 7.41·41-s − 7.65·43-s + 0.951·45-s − 10.1·47-s − 5.35·49-s + 1.31·51-s − 3.23·53-s + 0.904·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.425·5-s + 0.484·7-s + 0.333·9-s + 0.286·11-s − 0.784·13-s + 0.245·15-s + 0.318·17-s − 1.36·19-s + 0.279·21-s + 0.535·23-s − 0.819·25-s + 0.192·27-s − 0.462·29-s − 0.176·31-s + 0.165·33-s + 0.206·35-s − 1.28·37-s − 0.452·39-s − 1.15·41-s − 1.16·43-s + 0.141·45-s − 1.47·47-s − 0.765·49-s + 0.184·51-s − 0.444·53-s + 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.951T + 5T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 - 0.951T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 + 5.94T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 + 2.48T + 29T^{2} \) |
| 31 | \( 1 + 0.984T + 31T^{2} \) |
| 37 | \( 1 + 7.81T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 3.23T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 8.14T + 67T^{2} \) |
| 71 | \( 1 + 1.71T + 71T^{2} \) |
| 73 | \( 1 + 0.502T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 6.15T + 89T^{2} \) |
| 97 | \( 1 - 4.75T + 97T^{2} \) |
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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
−7.60406786388655057943490577030, −6.76878881648988193276437630079, −6.26928144934295778357853127347, −5.18625620149585209600657319471, −4.79243218081290810322459184763, −3.80454460852184534237914880659, −3.12351731201298354217245712912, −2.03437510958205305060843040834, −1.64281000035156101438619902158, 0,
1.64281000035156101438619902158, 2.03437510958205305060843040834, 3.12351731201298354217245712912, 3.80454460852184534237914880659, 4.79243218081290810322459184763, 5.18625620149585209600657319471, 6.26928144934295778357853127347, 6.76878881648988193276437630079, 7.60406786388655057943490577030
