L(s) = 1 | − 3·5-s + 11-s + 5·13-s + 6·17-s − 19-s + 4·25-s − 9·29-s − 8·31-s + 7·37-s + 6·41-s + 10·43-s + 3·47-s + 6·53-s − 3·55-s + 9·59-s + 2·61-s − 15·65-s + 13·67-s − 6·71-s − 11·73-s − 10·79-s + 6·83-s − 18·85-s + 6·89-s + 3·95-s + 10·97-s + 101-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.301·11-s + 1.38·13-s + 1.45·17-s − 0.229·19-s + 4/5·25-s − 1.67·29-s − 1.43·31-s + 1.15·37-s + 0.937·41-s + 1.52·43-s + 0.437·47-s + 0.824·53-s − 0.404·55-s + 1.17·59-s + 0.256·61-s − 1.86·65-s + 1.58·67-s − 0.712·71-s − 1.28·73-s − 1.12·79-s + 0.658·83-s − 1.95·85-s + 0.635·89-s + 0.307·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.84517221070759, −12.51466861748532, −11.78767082074552, −11.52953755388817, −11.16961016837239, −10.74386200852707, −10.25410050807570, −9.594133936347120, −9.115630867677814, −8.764002934963070, −8.196978442494666, −7.742214664649557, −7.354113289096674, −7.112201766709748, −6.145449426607868, −5.866901541939138, −5.464257532336432, −4.707931164741913, −3.999312578693823, −3.748166644545983, −3.562931539995704, −2.705140914239486, −2.070937674966941, −1.186138656633011, −0.8550058471875544, 0,
0.8550058471875544, 1.186138656633011, 2.070937674966941, 2.705140914239486, 3.562931539995704, 3.748166644545983, 3.999312578693823, 4.707931164741913, 5.464257532336432, 5.866901541939138, 6.145449426607868, 7.112201766709748, 7.354113289096674, 7.742214664649557, 8.196978442494666, 8.764002934963070, 9.115630867677814, 9.594133936347120, 10.25410050807570, 10.74386200852707, 11.16961016837239, 11.52953755388817, 11.78767082074552, 12.51466861748532, 12.84517221070759