L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 2·11-s − 12-s + 13-s + 16-s − 3·17-s + 18-s + 2·22-s − 23-s − 24-s + 26-s − 27-s − 5·29-s − 7·31-s + 32-s − 2·33-s − 3·34-s + 36-s − 2·37-s − 39-s − 7·41-s − 11·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.426·22-s − 0.208·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.928·29-s − 1.25·31-s + 0.176·32-s − 0.348·33-s − 0.514·34-s + 1/6·36-s − 0.328·37-s − 0.160·39-s − 1.09·41-s − 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.33732670856181311035087055551, −6.59265768992468053069122162408, −6.28381084691976507477185806090, −5.25674174038577756401080540033, −4.92885223889941429112242618891, −3.83695843790694712254078466884, −3.50365037746428250259417318781, −2.18519844757328419643733632491, −1.46480807728597688872197621704, 0,
1.46480807728597688872197621704, 2.18519844757328419643733632491, 3.50365037746428250259417318781, 3.83695843790694712254078466884, 4.92885223889941429112242618891, 5.25674174038577756401080540033, 6.28381084691976507477185806090, 6.59265768992468053069122162408, 7.33732670856181311035087055551