L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 11-s + 13-s + 15-s − 3·17-s + 2·19-s − 3·21-s − 5·23-s + 25-s + 27-s − 6·29-s − 10·31-s − 33-s − 3·35-s + 5·37-s + 39-s + 3·41-s − 4·43-s + 45-s − 6·47-s + 2·49-s − 3·51-s + 5·53-s − 55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s − 0.654·21-s − 1.04·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.79·31-s − 0.174·33-s − 0.507·35-s + 0.821·37-s + 0.160·39-s + 0.468·41-s − 0.609·43-s + 0.149·45-s − 0.875·47-s + 2/7·49-s − 0.420·51-s + 0.686·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−8.398716940790252906017435476279, −7.53519268379968514202282786762, −6.86202884976249809144758704348, −6.04834263172891330843858304757, −5.40769436447667749630416000395, −4.19175962408500603368543210021, −3.48485256207437635111202076201, −2.61153019287754941108481574107, −1.69203721417361487616392037040, 0,
1.69203721417361487616392037040, 2.61153019287754941108481574107, 3.48485256207437635111202076201, 4.19175962408500603368543210021, 5.40769436447667749630416000395, 6.04834263172891330843858304757, 6.86202884976249809144758704348, 7.53519268379968514202282786762, 8.398716940790252906017435476279