L(s) = 1 | − 3-s + 1.41·5-s + 1.41·7-s + 9-s − 1.41·15-s − 1.41·17-s − 1.41·19-s − 1.41·21-s + 23-s + 1.00·25-s − 27-s + 2.00·35-s + 1.41·43-s + 1.41·45-s + 1.00·49-s + 1.41·51-s − 1.41·53-s + 1.41·57-s + 1.41·63-s − 1.41·67-s − 69-s + 2·71-s − 2·73-s − 1.00·75-s − 1.41·79-s + 81-s − 2.00·85-s + ⋯ |
L(s) = 1 | − 3-s + 1.41·5-s + 1.41·7-s + 9-s − 1.41·15-s − 1.41·17-s − 1.41·19-s − 1.41·21-s + 23-s + 1.00·25-s − 27-s + 2.00·35-s + 1.41·43-s + 1.41·45-s + 1.00·49-s + 1.41·51-s − 1.41·53-s + 1.41·57-s + 1.41·63-s − 1.41·67-s − 69-s + 2·71-s − 2·73-s − 1.00·75-s − 1.41·79-s + 81-s − 2.00·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.087930335\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.087930335\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 + 2T + T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.25929506811972435766919237761, −9.256367422211388818371432824175, −8.569445135946363737269325911629, −7.38667988818145706654476376800, −6.47367438641744663291352348523, −5.84464475579063548732857058274, −4.89998424796880393431847506936, −4.38145930497602692576777313894, −2.32162399434700337058209002122, −1.50727428499267185578308857284,
1.50727428499267185578308857284, 2.32162399434700337058209002122, 4.38145930497602692576777313894, 4.89998424796880393431847506936, 5.84464475579063548732857058274, 6.47367438641744663291352348523, 7.38667988818145706654476376800, 8.569445135946363737269325911629, 9.256367422211388818371432824175, 10.25929506811972435766919237761