| L(s) = 1 | + 3-s − 1.61·5-s − 2.77·7-s + 9-s − 3.41·11-s − 6.91·13-s − 1.61·15-s + 3.58·17-s + 5.37·19-s − 2.77·21-s + 1.72·23-s − 2.39·25-s + 27-s − 7.25·29-s + 2.36·31-s − 3.41·33-s + 4.48·35-s − 5.21·37-s − 6.91·39-s − 1.85·41-s − 1.96·43-s − 1.61·45-s − 7.93·47-s + 0.720·49-s + 3.58·51-s + 0.173·53-s + 5.50·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.721·5-s − 1.05·7-s + 0.333·9-s − 1.02·11-s − 1.91·13-s − 0.416·15-s + 0.870·17-s + 1.23·19-s − 0.606·21-s + 0.358·23-s − 0.479·25-s + 0.192·27-s − 1.34·29-s + 0.424·31-s − 0.594·33-s + 0.757·35-s − 0.857·37-s − 1.10·39-s − 0.289·41-s − 0.299·43-s − 0.240·45-s − 1.15·47-s + 0.102·49-s + 0.502·51-s + 0.0238·53-s + 0.742·55-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
| \(L(1)\) |
\(\approx\) |
\(0.9275277241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9275277241\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 6.91T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 - 1.72T + 23T^{2} \) |
| 29 | \( 1 + 7.25T + 29T^{2} \) |
| 31 | \( 1 - 2.36T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 + 1.85T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 + 7.93T + 47T^{2} \) |
| 53 | \( 1 - 0.173T + 53T^{2} \) |
| 59 | \( 1 - 8.14T + 59T^{2} \) |
| 61 | \( 1 - 6.08T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 7.48T + 71T^{2} \) |
| 73 | \( 1 + 3.99T + 73T^{2} \) |
| 79 | \( 1 - 6.01T + 79T^{2} \) |
| 83 | \( 1 - 9.27T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−7.84121813868077850426705561333, −7.20448105107726058170141333886, −6.80519260820747043796800357920, −5.43929301494757769554836802564, −5.22152411109319089333028262292, −4.14049035241838347592109487880, −3.28862824898149453253140464664, −2.92441337861540244754509001782, −1.96610010391744008960858465316, −0.43458741052574529605558666826,
0.43458741052574529605558666826, 1.96610010391744008960858465316, 2.92441337861540244754509001782, 3.28862824898149453253140464664, 4.14049035241838347592109487880, 5.22152411109319089333028262292, 5.43929301494757769554836802564, 6.80519260820747043796800357920, 7.20448105107726058170141333886, 7.84121813868077850426705561333
