| L(s) = 1 | − 3-s − 3·5-s + 7-s − 2·9-s − 6·11-s + 13-s + 3·15-s − 3·17-s − 2·19-s − 21-s + 4·25-s + 5·27-s + 6·29-s + 4·31-s + 6·33-s − 3·35-s − 7·37-s − 39-s + 43-s + 6·45-s − 3·47-s − 6·49-s + 3·51-s + 18·55-s + 2·57-s + 6·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s − 1.80·11-s + 0.277·13-s + 0.774·15-s − 0.727·17-s − 0.458·19-s − 0.218·21-s + 4/5·25-s + 0.962·27-s + 1.11·29-s + 0.718·31-s + 1.04·33-s − 0.507·35-s − 1.15·37-s − 0.160·39-s + 0.152·43-s + 0.894·45-s − 0.437·47-s − 6/7·49-s + 0.420·51-s + 2.42·55-s + 0.264·57-s + 0.781·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 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
−11.76197823031083275943388088877, −11.05532024538312794409968370648, −10.33835351049427822769308975614, −8.499286317189647023036962817405, −8.059393957478115811114793708747, −6.80992153386544904186537940607, −5.41734923280427435562979846122, −4.44297288531505612446396437446, −2.86140175572535291493434732878, 0,
2.86140175572535291493434732878, 4.44297288531505612446396437446, 5.41734923280427435562979846122, 6.80992153386544904186537940607, 8.059393957478115811114793708747, 8.499286317189647023036962817405, 10.33835351049427822769308975614, 11.05532024538312794409968370648, 11.76197823031083275943388088877
