| L(s) = 1 | − 2·3-s + 5-s + 4·7-s + 9-s + 4·13-s − 2·15-s + 4·19-s − 8·21-s − 6·23-s + 25-s + 4·27-s + 6·29-s + 8·31-s + 4·35-s + 2·37-s − 8·39-s − 6·41-s − 8·43-s + 45-s + 6·47-s + 9·49-s − 6·53-s − 8·57-s − 12·59-s − 2·61-s + 4·63-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.10·13-s − 0.516·15-s + 0.917·19-s − 1.74·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.676·35-s + 0.328·37-s − 1.28·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.05·57-s − 1.56·59-s − 0.256·61-s + 0.503·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.666638438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666638438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.760919737093886217653898626233, −8.247303296581138836487669927099, −7.43121115888886915787668656338, −6.23030064104064618549044875670, −6.00141492759498146682778993334, −4.92794271190861894483214793107, −4.58917197940168603932789584872, −3.20812926244335147042387019020, −1.81835163843956709721896091480, −0.945394974935148940289301728115,
0.945394974935148940289301728115, 1.81835163843956709721896091480, 3.20812926244335147042387019020, 4.58917197940168603932789584872, 4.92794271190861894483214793107, 6.00141492759498146682778993334, 6.23030064104064618549044875670, 7.43121115888886915787668656338, 8.247303296581138836487669927099, 8.760919737093886217653898626233
