L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 3·9-s − 10-s − 6·13-s + 14-s + 16-s − 2·17-s − 3·18-s − 4·19-s − 20-s − 23-s + 25-s − 6·26-s + 28-s − 2·29-s − 8·31-s + 32-s − 2·34-s − 35-s − 3·36-s + 10·37-s − 4·38-s − 40-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.917·19-s − 0.223·20-s − 0.208·23-s + 1/5·25-s − 1.17·26-s + 0.188·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s − 1/2·36-s + 1.64·37-s − 0.648·38-s − 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.946453471903430102336697144102, −8.046129644320038588201654210362, −7.40948143654137681857974983722, −6.50865843871373221304541887022, −5.57811808978452044651121051260, −4.81757800310524226250297746310, −4.03838100423666482331336280000, −2.88110259535060963772124936999, −2.06974061265330760784395788256, 0,
2.06974061265330760784395788256, 2.88110259535060963772124936999, 4.03838100423666482331336280000, 4.81757800310524226250297746310, 5.57811808978452044651121051260, 6.50865843871373221304541887022, 7.40948143654137681857974983722, 8.046129644320038588201654210362, 8.946453471903430102336697144102