L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s + 7-s + 3·8-s + 9-s − 10-s + 3·11-s + 2·12-s − 7·13-s − 14-s − 2·15-s − 16-s + 6·17-s − 18-s + 4·19-s − 20-s − 2·21-s − 3·22-s − 6·24-s + 25-s + 7·26-s + 4·27-s − 28-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.577·12-s − 1.94·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.436·21-s − 0.639·22-s − 1.22·24-s + 1/5·25-s + 1.37·26-s + 0.769·27-s − 0.188·28-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 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
−7.37106214057316294883844223891, −7.05464721443109212489663336571, −5.94730756673747772852985893641, −5.27888575519341351589297354655, −5.01996859599546007100040383058, −4.10940944328384169042465976657, −3.06785834291586159922919521518, −1.79926099470391730437965836740, −1.01522590757540824247174525512, 0,
1.01522590757540824247174525512, 1.79926099470391730437965836740, 3.06785834291586159922919521518, 4.10940944328384169042465976657, 5.01996859599546007100040383058, 5.27888575519341351589297354655, 5.94730756673747772852985893641, 7.05464721443109212489663336571, 7.37106214057316294883844223891