| L(s) = 1 | − 1.53·2-s − 3-s + 1.34·4-s + 1.53·6-s + 0.347·7-s − 0.532·8-s + 9-s + 1.87·11-s − 1.34·12-s + 1.53·13-s − 0.532·14-s − 0.532·16-s + 1.87·17-s − 1.53·18-s − 0.347·21-s − 2.87·22-s + 0.532·24-s − 2.34·26-s − 27-s + 0.467·28-s − 29-s + 1.34·32-s − 1.87·33-s − 2.87·34-s + 1.34·36-s − 1.53·39-s + 41-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 3-s + 1.34·4-s + 1.53·6-s + 0.347·7-s − 0.532·8-s + 9-s + 1.87·11-s − 1.34·12-s + 1.53·13-s − 0.532·14-s − 0.532·16-s + 1.87·17-s − 1.53·18-s − 0.347·21-s − 2.87·22-s + 0.532·24-s − 2.34·26-s − 27-s + 0.467·28-s − 29-s + 1.34·32-s − 1.87·33-s − 2.87·34-s + 1.34·36-s − 1.53·39-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5531089543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5531089543\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + T^{2} \) |
| 7 | \( 1 - 0.347T + T^{2} \) |
| 11 | \( 1 - 1.87T + T^{2} \) |
| 13 | \( 1 - 1.53T + T^{2} \) |
| 17 | \( 1 - 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 0.347T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.53T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.370231798785217174267696211616, −8.579943281937858846332289848704, −7.78568402852589979850302836332, −7.06299408771861551595904807137, −6.25909602560367724146302844124, −5.67830604541890026810108894041, −4.35847235339394158100138240385, −3.52757895222731274896147097187, −1.51489068005775694053626102262, −1.16696546040772382780751521796,
1.16696546040772382780751521796, 1.51489068005775694053626102262, 3.52757895222731274896147097187, 4.35847235339394158100138240385, 5.67830604541890026810108894041, 6.25909602560367724146302844124, 7.06299408771861551595904807137, 7.78568402852589979850302836332, 8.579943281937858846332289848704, 9.370231798785217174267696211616
