| L(s) = 1 | + 2.54·3-s + 1.59·5-s − 2.13·7-s + 3.46·9-s − 5.04·11-s − 1.30·13-s + 4.06·15-s + 1.38·17-s − 3.89·19-s − 5.42·21-s − 7.19·23-s − 2.44·25-s + 1.16·27-s − 3.67·29-s − 1.32·31-s − 12.8·33-s − 3.41·35-s − 1.51·37-s − 3.31·39-s − 8.98·41-s − 6.23·43-s + 5.52·45-s + 6.82·47-s − 2.44·49-s + 3.52·51-s + 2.91·53-s − 8.06·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 0.714·5-s − 0.806·7-s + 1.15·9-s − 1.52·11-s − 0.361·13-s + 1.04·15-s + 0.336·17-s − 0.893·19-s − 1.18·21-s − 1.50·23-s − 0.489·25-s + 0.225·27-s − 0.681·29-s − 0.238·31-s − 2.23·33-s − 0.576·35-s − 0.249·37-s − 0.530·39-s − 1.40·41-s − 0.950·43-s + 0.824·45-s + 0.995·47-s − 0.349·49-s + 0.493·51-s + 0.399·53-s − 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 + 5.04T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 + 7.19T + 23T^{2} \) |
| 29 | \( 1 + 3.67T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 - 2.91T + 53T^{2} \) |
| 59 | \( 1 - 9.42T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 0.381T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.64T + 79T^{2} \) |
| 83 | \( 1 + 0.795T + 83T^{2} \) |
| 89 | \( 1 + 0.551T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−8.179746603102184106840367548219, −7.55479264581376814591567866420, −6.72996886990396165367172582286, −5.83043156582156109347005005201, −5.15232172631541867097094233683, −3.93886313333849881544771497411, −3.31924779449801116938154375028, −2.31635925449867468372808163276, −2.04544835812598832916041014582, 0,
2.04544835812598832916041014582, 2.31635925449867468372808163276, 3.31924779449801116938154375028, 3.93886313333849881544771497411, 5.15232172631541867097094233683, 5.83043156582156109347005005201, 6.72996886990396165367172582286, 7.55479264581376814591567866420, 8.179746603102184106840367548219
