L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 12-s − 2·13-s + 14-s + 16-s − 6·17-s − 2·18-s − 19-s + 21-s − 9·23-s + 24-s − 5·25-s − 2·26-s − 5·27-s + 28-s − 3·29-s + 8·31-s + 32-s − 6·34-s − 2·36-s − 7·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.471·18-s − 0.229·19-s + 0.218·21-s − 1.87·23-s + 0.204·24-s − 25-s − 0.392·26-s − 0.962·27-s + 0.188·28-s − 0.557·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 1/3·36-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.899739217449880975374081463535, −7.35206448560444073899063892559, −6.22159707637516768369011293624, −5.91546062603574057521834834184, −4.78797471012671294437139039244, −4.24321369702620947143076158240, −3.38754182552549129939963338719, −2.38660471418585902147181592994, −1.92463430074466681468166332995, 0,
1.92463430074466681468166332995, 2.38660471418585902147181592994, 3.38754182552549129939963338719, 4.24321369702620947143076158240, 4.78797471012671294437139039244, 5.91546062603574057521834834184, 6.22159707637516768369011293624, 7.35206448560444073899063892559, 7.899739217449880975374081463535