L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 11-s − 12-s + 13-s − 14-s + 16-s − 6·17-s + 18-s + 21-s − 22-s − 24-s + 26-s − 27-s − 28-s + 6·29-s + 31-s + 32-s + 33-s − 6·34-s + 36-s + 5·37-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 + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.218·21-s − 0.213·22-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.179·31-s + 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s + 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.84204840906848359884327486965, −6.90133229921728365535751443044, −6.42189291458458688505511587086, −5.83366032476747166300676798636, −4.80327014984321196217000035389, −4.46007839354697637445341339047, −3.38436850756224246282310830134, −2.57482715218498483234227841295, −1.47462250124641680907921594200, 0,
1.47462250124641680907921594200, 2.57482715218498483234227841295, 3.38436850756224246282310830134, 4.46007839354697637445341339047, 4.80327014984321196217000035389, 5.83366032476747166300676798636, 6.42189291458458688505511587086, 6.90133229921728365535751443044, 7.84204840906848359884327486965