| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2580621681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2580621681\) |
| \(L(1)\) |
|
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
−9.609377938795900997465534754707, −8.348698532668616590402293614163, −7.68448898919329146098893357109, −7.28625155324289672698272817718, −6.33749050401458408994139787064, −5.26844723531087514107282166394, −4.83731914147215123808324809836, −3.13302973908668216888296534943, −2.06303999472383132311104479016, −0.62600961268415776705364245821,
0.62600961268415776705364245821, 2.06303999472383132311104479016, 3.13302973908668216888296534943, 4.83731914147215123808324809836, 5.26844723531087514107282166394, 6.33749050401458408994139787064, 7.28625155324289672698272817718, 7.68448898919329146098893357109, 8.348698532668616590402293614163, 9.609377938795900997465534754707
