L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 11-s − 12-s − 4·13-s − 16-s − 3·17-s + 18-s − 19-s − 22-s − 23-s − 3·24-s − 5·25-s − 4·26-s + 27-s − 5·29-s + 10·31-s + 5·32-s − 33-s − 3·34-s − 36-s − 11·37-s − 38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.10·13-s − 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.229·19-s − 0.213·22-s − 0.208·23-s − 0.612·24-s − 25-s − 0.784·26-s + 0.192·27-s − 0.928·29-s + 1.79·31-s + 0.883·32-s − 0.174·33-s − 0.514·34-s − 1/6·36-s − 1.80·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.890218680689816594150212632257, −8.348070892230668419090874555738, −7.39609673319231247420248676317, −6.52713736177357209970737811683, −5.48804579558673623840334833320, −4.74643451745337726581531528707, −3.97487393053003849873987102843, −3.03060060109820558625370250970, −2.04564581170762993293954477416, 0,
2.04564581170762993293954477416, 3.03060060109820558625370250970, 3.97487393053003849873987102843, 4.74643451745337726581531528707, 5.48804579558673623840334833320, 6.52713736177357209970737811683, 7.39609673319231247420248676317, 8.348070892230668419090874555738, 8.890218680689816594150212632257