L(s) = 1 | − 2·4-s − 5-s + 13-s + 4·16-s + 6·17-s − 5·19-s + 2·20-s − 6·23-s + 25-s + 6·29-s − 5·31-s − 7·37-s + 12·41-s − 43-s + 6·47-s − 2·52-s − 6·59-s − 2·61-s − 8·64-s − 65-s − 7·67-s − 12·68-s − 12·71-s − 11·73-s + 10·76-s − 13·79-s − 4·80-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s + 0.277·13-s + 16-s + 1.45·17-s − 1.14·19-s + 0.447·20-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.898·31-s − 1.15·37-s + 1.87·41-s − 0.152·43-s + 0.875·47-s − 0.277·52-s − 0.781·59-s − 0.256·61-s − 64-s − 0.124·65-s − 0.855·67-s − 1.45·68-s − 1.42·71-s − 1.28·73-s + 1.14·76-s − 1.46·79-s − 0.447·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.660591871742027100593044542530, −7.986194738400219607516703979848, −7.34208353369648062971602623315, −6.11199301680000687274985177213, −5.52422240585261813833033374684, −4.43197376376442028675221832458, −3.92183284053267882397615911864, −2.91971767189759948196763143522, −1.37971581701681221015186904288, 0,
1.37971581701681221015186904288, 2.91971767189759948196763143522, 3.92183284053267882397615911864, 4.43197376376442028675221832458, 5.52422240585261813833033374684, 6.11199301680000687274985177213, 7.34208353369648062971602623315, 7.986194738400219607516703979848, 8.660591871742027100593044542530