L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s − 2·9-s − 6·11-s − 12-s − 5·13-s + 16-s − 3·17-s + 2·18-s − 19-s + 6·22-s + 3·23-s + 24-s − 5·25-s + 5·26-s + 5·27-s + 9·29-s + 4·31-s − 32-s + 6·33-s + 3·34-s − 2·36-s + 2·37-s + 38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s − 2/3·9-s − 1.80·11-s − 0.288·12-s − 1.38·13-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.229·19-s + 1.27·22-s + 0.625·23-s + 0.204·24-s − 25-s + 0.980·26-s + 0.962·27-s + 1.67·29-s + 0.718·31-s − 0.176·32-s + 1.04·33-s + 0.514·34-s − 1/3·36-s + 0.328·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4545505667\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4545505667\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.291172104782228986263782930604, −8.324937534107450862989290709661, −7.82053034531950964944245413219, −6.95287787696203506152668709557, −6.09183089055341806901121983872, −5.23619780971878042674541982146, −4.59019380456957647836499666225, −2.84763354584267834606402815861, −2.37108201360933257056349110850, −0.48486995001118461060091950560,
0.48486995001118461060091950560, 2.37108201360933257056349110850, 2.84763354584267834606402815861, 4.59019380456957647836499666225, 5.23619780971878042674541982146, 6.09183089055341806901121983872, 6.95287787696203506152668709557, 7.82053034531950964944245413219, 8.324937534107450862989290709661, 9.291172104782228986263782930604