| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 6·11-s − 4·13-s − 15-s + 6·17-s + 2·19-s + 21-s + 25-s + 27-s + 6·29-s − 10·31-s + 6·33-s − 35-s + 2·37-s − 4·39-s − 6·41-s − 4·43-s − 45-s + 49-s + 6·51-s − 12·53-s − 6·55-s + 2·57-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.258·15-s + 1.45·17-s + 0.458·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s + 1.04·33-s − 0.169·35-s + 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s − 1.64·53-s − 0.809·55-s + 0.264·57-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.728121110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728121110\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−11.41468963687341383833066905661, −10.07251256598781585775622826482, −9.402208417888729715176696972987, −8.449326916407346032985487608845, −7.52869725497309556858557046810, −6.74469025095782946469111005872, −5.31154804232921291759313294728, −4.14525984671534409011710586104, −3.16591912200898921503339464095, −1.47447091118007102923277723016,
1.47447091118007102923277723016, 3.16591912200898921503339464095, 4.14525984671534409011710586104, 5.31154804232921291759313294728, 6.74469025095782946469111005872, 7.52869725497309556858557046810, 8.449326916407346032985487608845, 9.402208417888729715176696972987, 10.07251256598781585775622826482, 11.41468963687341383833066905661
