| L(s) = 1 | − 3-s − 5-s − 5·7-s + 9-s + 4·13-s + 15-s − 3·17-s − 4·19-s + 5·21-s − 23-s + 25-s − 27-s + 29-s + 31-s + 5·35-s − 37-s − 4·39-s + 11·41-s + 4·43-s − 45-s + 6·47-s + 18·49-s + 3·51-s + 53-s + 4·57-s − 59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 1.10·13-s + 0.258·15-s − 0.727·17-s − 0.917·19-s + 1.09·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.179·31-s + 0.845·35-s − 0.164·37-s − 0.640·39-s + 1.71·41-s + 0.609·43-s − 0.149·45-s + 0.875·47-s + 18/7·49-s + 0.420·51-s + 0.137·53-s + 0.529·57-s − 0.130·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7833992457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7833992457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.490834483076530112950594729522, −8.972782544307586558194185458056, −7.939682391816145384225408893687, −6.82030863704119938120900900769, −6.38765866408695998791678634939, −5.66910598157464769461844955537, −4.26457759009902094548130755028, −3.65815898084270096861369071239, −2.50345116392380435330633055071, −0.63577611505595346610175523130,
0.63577611505595346610175523130, 2.50345116392380435330633055071, 3.65815898084270096861369071239, 4.26457759009902094548130755028, 5.66910598157464769461844955537, 6.38765866408695998791678634939, 6.82030863704119938120900900769, 7.939682391816145384225408893687, 8.972782544307586558194185458056, 9.490834483076530112950594729522
