L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s + 4·13-s − 15-s + 6·17-s + 7·19-s − 21-s − 23-s + 25-s − 27-s − 6·29-s + 4·31-s + 33-s + 35-s − 2·37-s − 4·39-s + 9·41-s + 2·43-s + 45-s + 7·47-s + 49-s − 6·51-s + 5·53-s − 55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s + 1.60·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s + 0.169·35-s − 0.328·37-s − 0.640·39-s + 1.40·41-s + 0.304·43-s + 0.149·45-s + 1.02·47-s + 1/7·49-s − 0.840·51-s + 0.686·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.472630130\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.472630130\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.64222071249777309420162100265, −7.07202991726494029092480482092, −6.07248405517259264181598087947, −5.62467582435781728943707867361, −5.21786473129594586744537208960, −4.18749598134104065704818762638, −3.48488895114235697623238538780, −2.61463233742959485888326091410, −1.45346245589439546542862235463, −0.873591127470956303153087243496,
0.873591127470956303153087243496, 1.45346245589439546542862235463, 2.61463233742959485888326091410, 3.48488895114235697623238538780, 4.18749598134104065704818762638, 5.21786473129594586744537208960, 5.62467582435781728943707867361, 6.07248405517259264181598087947, 7.07202991726494029092480482092, 7.64222071249777309420162100265