L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s − 6·13-s + 2·15-s − 2·17-s − 6·19-s − 4·21-s + 4·23-s + 25-s − 4·27-s − 2·29-s + 6·31-s − 2·35-s + 37-s − 12·39-s − 6·41-s − 8·43-s + 45-s + 6·47-s − 3·49-s − 4·51-s − 10·53-s − 12·57-s − 14·59-s + 6·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.66·13-s + 0.516·15-s − 0.485·17-s − 1.37·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.769·27-s − 0.371·29-s + 1.07·31-s − 0.338·35-s + 0.164·37-s − 1.92·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.560·51-s − 1.37·53-s − 1.58·57-s − 1.82·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.459643001244321750078091777712, −7.73321943834775044441688792893, −6.85310964676830604987865802422, −6.31470013681956299359468304822, −5.14999863116616586261451311739, −4.39736167147443531373968012838, −3.28202572581050957428380625087, −2.63827196435325679302167554496, −1.91068492214473856215587510083, 0,
1.91068492214473856215587510083, 2.63827196435325679302167554496, 3.28202572581050957428380625087, 4.39736167147443531373968012838, 5.14999863116616586261451311739, 6.31470013681956299359468304822, 6.85310964676830604987865802422, 7.73321943834775044441688792893, 8.459643001244321750078091777712