| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s + 9-s − 11-s + 12-s + 13-s + 3·14-s + 16-s + 17-s + 18-s − 6·19-s + 3·21-s − 22-s − 5·23-s + 24-s + 26-s + 27-s + 3·28-s + 5·29-s − 9·31-s + 32-s − 33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.37·19-s + 0.654·21-s − 0.213·22-s − 1.04·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.566·28-s + 0.928·29-s − 1.61·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.89884133210319, −12.23298512164749, −12.05341616823804, −11.19902984237897, −11.07377809731102, −10.68994035749214, −10.03062030216048, −9.623123526213068, −9.048156737311360, −8.467515143552375, −8.053107421469293, −7.849302453827411, −7.299735397332345, −6.656077342907174, −6.152643098223007, −5.798989580189492, −5.145484201283027, −4.477772287100935, −4.457476463566635, −3.756287006956518, −3.213462280763892, −2.595541740823588, −2.029025959896867, −1.709404255927012, −0.9804218664183033, 0,
0.9804218664183033, 1.709404255927012, 2.029025959896867, 2.595541740823588, 3.213462280763892, 3.756287006956518, 4.457476463566635, 4.477772287100935, 5.145484201283027, 5.798989580189492, 6.152643098223007, 6.656077342907174, 7.299735397332345, 7.849302453827411, 8.053107421469293, 8.467515143552375, 9.048156737311360, 9.623123526213068, 10.03062030216048, 10.68994035749214, 11.07377809731102, 11.19902984237897, 12.05341616823804, 12.23298512164749, 12.89884133210319
