L(s) = 1 | − 3-s + 9-s − 4·13-s − 4·17-s + 4·19-s + 4·23-s − 5·25-s − 27-s + 2·29-s − 8·31-s − 6·37-s + 4·39-s − 12·41-s + 4·43-s + 8·47-s + 4·51-s + 6·53-s − 4·57-s − 12·59-s − 4·61-s − 4·67-s − 4·69-s − 12·71-s − 8·73-s + 5·75-s − 16·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.10·13-s − 0.970·17-s + 0.917·19-s + 0.834·23-s − 25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.640·39-s − 1.87·41-s + 0.609·43-s + 1.16·47-s + 0.560·51-s + 0.824·53-s − 0.529·57-s − 1.56·59-s − 0.512·61-s − 0.488·67-s − 0.481·69-s − 1.42·71-s − 0.936·73-s + 0.577·75-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−9.402598807518549292554775867883, −8.679980472024348350693305920647, −7.39368277172255819158994073993, −7.06575029892920470972632576297, −5.87340118343345405222811842346, −5.13242773052280229930287694957, −4.26842049846374598919859794951, −3.02749641502338293168200065073, −1.73728899486802098617607613127, 0,
1.73728899486802098617607613127, 3.02749641502338293168200065073, 4.26842049846374598919859794951, 5.13242773052280229930287694957, 5.87340118343345405222811842346, 7.06575029892920470972632576297, 7.39368277172255819158994073993, 8.679980472024348350693305920647, 9.402598807518549292554775867883