| L(s) = 1 | − 0.823·3-s − 2.12·5-s − 4.15·7-s − 2.32·9-s − 0.347·11-s + 3.83·13-s + 1.74·15-s − 6.72·17-s + 6.76·19-s + 3.42·21-s − 0.816·23-s − 0.494·25-s + 4.38·27-s + 1.71·29-s − 2.32·31-s + 0.285·33-s + 8.81·35-s + 5.48·37-s − 3.16·39-s − 1.70·41-s + 5.51·43-s + 4.92·45-s + 6.07·47-s + 10.2·49-s + 5.54·51-s + 3.81·53-s + 0.737·55-s + ⋯ |
| L(s) = 1 | − 0.475·3-s − 0.949·5-s − 1.57·7-s − 0.773·9-s − 0.104·11-s + 1.06·13-s + 0.451·15-s − 1.63·17-s + 1.55·19-s + 0.746·21-s − 0.170·23-s − 0.0988·25-s + 0.843·27-s + 0.318·29-s − 0.418·31-s + 0.0497·33-s + 1.49·35-s + 0.902·37-s − 0.506·39-s − 0.266·41-s + 0.840·43-s + 0.734·45-s + 0.885·47-s + 1.46·49-s + 0.775·51-s + 0.524·53-s + 0.0993·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.823T + 3T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 11 | \( 1 + 0.347T + 11T^{2} \) |
| 13 | \( 1 - 3.83T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 - 6.76T + 19T^{2} \) |
| 23 | \( 1 + 0.816T + 23T^{2} \) |
| 29 | \( 1 - 1.71T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 - 5.48T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 - 6.07T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 - 8.10T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 - 7.22T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 4.46T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 1.00T + 83T^{2} \) |
| 89 | \( 1 + 8.30T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.35519761794097744424899880402, −6.76620950860045877798818031272, −6.05781126158527216271558008820, −5.64576590655301005515091464901, −4.55922776483689132792349542555, −3.80535493985572714622088264184, −3.22460940654024536403958030398, −2.46308905071911455893621312274, −0.850138043771136430203071057853, 0,
0.850138043771136430203071057853, 2.46308905071911455893621312274, 3.22460940654024536403958030398, 3.80535493985572714622088264184, 4.55922776483689132792349542555, 5.64576590655301005515091464901, 6.05781126158527216271558008820, 6.76620950860045877798818031272, 7.35519761794097744424899880402
