L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 8-s + 9-s − 2·10-s − 6·11-s − 2·12-s + 4·13-s − 4·15-s + 16-s + 2·17-s − 18-s − 2·19-s + 2·20-s + 6·22-s + 2·24-s − 25-s − 4·26-s + 4·27-s + 4·30-s − 32-s + 12·33-s − 2·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.80·11-s − 0.577·12-s + 1.10·13-s − 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.458·19-s + 0.447·20-s + 1.27·22-s + 0.408·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.730·30-s − 0.176·32-s + 2.08·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.042812083292086273503925835595, −7.45914327788800647810167327936, −6.27362296030827815619381860998, −6.02515845875248382835555308740, −5.38148102787186741958331923863, −4.55865301925676721492053108177, −3.14695056260572958526567132831, −2.27898879076152751798560422463, −1.16792656604651297935005555026, 0,
1.16792656604651297935005555026, 2.27898879076152751798560422463, 3.14695056260572958526567132831, 4.55865301925676721492053108177, 5.38148102787186741958331923863, 6.02515845875248382835555308740, 6.27362296030827815619381860998, 7.45914327788800647810167327936, 8.042812083292086273503925835595