L(s) = 1 | + 3·3-s + 32·7-s + 9·9-s + 60·11-s + 34·13-s − 42·17-s + 76·19-s + 96·21-s + 27·27-s + 6·29-s + 232·31-s + 180·33-s − 134·37-s + 102·39-s + 234·41-s − 412·43-s − 360·47-s + 681·49-s − 126·51-s − 222·53-s + 228·57-s − 660·59-s − 490·61-s + 288·63-s + 812·67-s − 120·71-s − 746·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.72·7-s + 1/3·9-s + 1.64·11-s + 0.725·13-s − 0.599·17-s + 0.917·19-s + 0.997·21-s + 0.192·27-s + 0.0384·29-s + 1.34·31-s + 0.949·33-s − 0.595·37-s + 0.418·39-s + 0.891·41-s − 1.46·43-s − 1.11·47-s + 1.98·49-s − 0.345·51-s − 0.575·53-s + 0.529·57-s − 1.45·59-s − 1.02·61-s + 0.575·63-s + 1.48·67-s − 0.200·71-s − 1.19·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.118714546\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.118714546\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 232 T + p^{3} T^{2} \) |
| 37 | \( 1 + 134 T + p^{3} T^{2} \) |
| 41 | \( 1 - 234 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 360 T + p^{3} T^{2} \) |
| 53 | \( 1 + 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 490 T + p^{3} T^{2} \) |
| 67 | \( 1 - 812 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 746 T + p^{3} T^{2} \) |
| 79 | \( 1 + 152 T + p^{3} T^{2} \) |
| 83 | \( 1 + 804 T + p^{3} T^{2} \) |
| 89 | \( 1 + 678 T + p^{3} T^{2} \) |
| 97 | \( 1 + 2 p T + p^{3} T^{2} \) |
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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
−9.164858327843259744665934240153, −8.545111329440150688332317992300, −7.927032409131370979175356898085, −6.97804788593288040156666683426, −6.10161346980439707981858285442, −4.86444372262751268927006658512, −4.23941902522525956850157665074, −3.19752820355494074183114053304, −1.76353162289035314921190208974, −1.18780418835045452296556171580,
1.18780418835045452296556171580, 1.76353162289035314921190208974, 3.19752820355494074183114053304, 4.23941902522525956850157665074, 4.86444372262751268927006658512, 6.10161346980439707981858285442, 6.97804788593288040156666683426, 7.927032409131370979175356898085, 8.545111329440150688332317992300, 9.164858327843259744665934240153