L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s − 4·11-s − 6·13-s + 2·14-s − 16-s − 4·17-s + 2·19-s + 4·22-s + 23-s − 5·25-s + 6·26-s + 2·28-s − 2·29-s + 4·31-s − 5·32-s + 4·34-s + 2·37-s − 2·38-s − 2·41-s + 10·43-s + 4·44-s − 46-s − 3·49-s + 5·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 1.20·11-s − 1.66·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 0.458·19-s + 0.852·22-s + 0.208·23-s − 25-s + 1.17·26-s + 0.377·28-s − 0.371·29-s + 0.718·31-s − 0.883·32-s + 0.685·34-s + 0.328·37-s − 0.324·38-s − 0.312·41-s + 1.52·43-s + 0.603·44-s − 0.147·46-s − 3/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.92095451813987707590181289229, −10.55178428658851003948359839715, −9.869974969451310136284054730415, −9.110362542146022510416950436672, −7.901424653634420272482508378615, −7.12753482747278279165710993198, −5.49966099869381914005238031655, −4.37441808392513917021094896393, −2.56565233054647425716584327864, 0,
2.56565233054647425716584327864, 4.37441808392513917021094896393, 5.49966099869381914005238031655, 7.12753482747278279165710993198, 7.901424653634420272482508378615, 9.110362542146022510416950436672, 9.869974969451310136284054730415, 10.55178428658851003948359839715, 11.92095451813987707590181289229