| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 3·13-s + 14-s + 16-s − 7·17-s + 6·19-s + 9·23-s + 3·26-s + 28-s + 3·29-s − 7·31-s + 32-s − 7·34-s − 10·37-s + 6·38-s − 41-s − 13·43-s + 9·46-s − 2·47-s + 49-s + 3·52-s − 53-s + 56-s + 3·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s + 1.37·19-s + 1.87·23-s + 0.588·26-s + 0.188·28-s + 0.557·29-s − 1.25·31-s + 0.176·32-s − 1.20·34-s − 1.64·37-s + 0.973·38-s − 0.156·41-s − 1.98·43-s + 1.32·46-s − 0.291·47-s + 1/7·49-s + 0.416·52-s − 0.137·53-s + 0.133·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.75194297717421, −12.30856355810895, −11.62604838362833, −11.48277929293318, −10.99984709121804, −10.62732700034943, −10.13575162777313, −9.504349387703694, −8.891103089772919, −8.705371070535528, −8.248570688532794, −7.399550970461192, −7.114284548418003, −6.789578316749081, −6.202630507470350, −5.617387373193845, −5.179732931628620, −4.723638797303165, −4.399885728884225, −3.439902937590306, −3.385025885308251, −2.770549682935796, −1.904679131149870, −1.609474970539989, −0.9008139753218012, 0,
0.9008139753218012, 1.609474970539989, 1.904679131149870, 2.770549682935796, 3.385025885308251, 3.439902937590306, 4.399885728884225, 4.723638797303165, 5.179732931628620, 5.617387373193845, 6.202630507470350, 6.789578316749081, 7.114284548418003, 7.399550970461192, 8.248570688532794, 8.705371070535528, 8.891103089772919, 9.504349387703694, 10.13575162777313, 10.62732700034943, 10.99984709121804, 11.48277929293318, 11.62604838362833, 12.30856355810895, 12.75194297717421
