| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 13-s − 15-s − 6·17-s + 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·35-s − 2·37-s + 39-s − 10·41-s + 4·43-s + 45-s + 8·47-s + 9·49-s + 6·51-s − 2·53-s + 4·59-s − 14·61-s − 4·63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 1.45·17-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s + 0.520·59-s − 1.79·61-s − 0.503·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.43017074385745, −12.76386204105567, −12.49961014960687, −12.00372262500330, −11.59040737666125, −10.75722245054978, −10.58473601464113, −10.13135848934364, −9.560703930579517, −9.140838371647045, −8.825477529012664, −8.120071853049097, −7.364378639506989, −6.992821430170514, −6.507165611147986, −6.124271962929566, −5.716101011898569, −5.049432544377119, −4.495205791953018, −3.962885568679200, −3.335513702732330, −2.781899499359334, −2.136374092432314, −1.583165940716244, −0.5476856088506360, 0,
0.5476856088506360, 1.583165940716244, 2.136374092432314, 2.781899499359334, 3.335513702732330, 3.962885568679200, 4.495205791953018, 5.049432544377119, 5.716101011898569, 6.124271962929566, 6.507165611147986, 6.992821430170514, 7.364378639506989, 8.120071853049097, 8.825477529012664, 9.140838371647045, 9.560703930579517, 10.13135848934364, 10.58473601464113, 10.75722245054978, 11.59040737666125, 12.00372262500330, 12.49961014960687, 12.76386204105567, 13.43017074385745
