| L(s) = 1 | − 3-s + 9-s − 2·13-s + 4·17-s − 2·19-s − 5·25-s − 27-s + 4·29-s − 2·31-s − 37-s + 2·39-s − 10·41-s + 2·43-s − 8·47-s − 7·49-s − 4·51-s + 10·53-s + 2·57-s + 10·61-s − 8·67-s + 16·71-s − 10·73-s + 5·75-s + 14·79-s + 81-s − 4·83-s − 4·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.554·13-s + 0.970·17-s − 0.458·19-s − 25-s − 0.192·27-s + 0.742·29-s − 0.359·31-s − 0.164·37-s + 0.320·39-s − 1.56·41-s + 0.304·43-s − 1.16·47-s − 49-s − 0.560·51-s + 1.37·53-s + 0.264·57-s + 1.28·61-s − 0.977·67-s + 1.89·71-s − 1.17·73-s + 0.577·75-s + 1.57·79-s + 1/9·81-s − 0.439·83-s − 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.82808907325012, −14.08306876931157, −13.78697060320961, −12.97240859045155, −12.70700929326977, −12.06103471461144, −11.62381276565004, −11.30026495589580, −10.42580372946552, −10.08474586192045, −9.774029845667730, −8.975754708778776, −8.399138758020888, −7.854965341252230, −7.309269013113775, −6.721665381531749, −6.194607605555644, −5.595228352309107, −5.026333059473357, −4.587986456468585, −3.707818395553251, −3.309473195623645, −2.329961569458402, −1.748895697888274, −0.8524818981865994, 0,
0.8524818981865994, 1.748895697888274, 2.329961569458402, 3.309473195623645, 3.707818395553251, 4.587986456468585, 5.026333059473357, 5.595228352309107, 6.194607605555644, 6.721665381531749, 7.309269013113775, 7.854965341252230, 8.399138758020888, 8.975754708778776, 9.774029845667730, 10.08474586192045, 10.42580372946552, 11.30026495589580, 11.62381276565004, 12.06103471461144, 12.70700929326977, 12.97240859045155, 13.78697060320961, 14.08306876931157, 14.82808907325012
