L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 3·11-s + 12-s − 13-s + 16-s − 2·17-s − 18-s + 19-s + 3·22-s − 9·23-s − 24-s + 26-s + 27-s − 10·29-s − 7·31-s − 32-s − 3·33-s + 2·34-s + 36-s + 2·37-s − 38-s − 39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.639·22-s − 1.87·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 1.85·29-s − 1.25·31-s − 0.176·32-s − 0.522·33-s + 0.342·34-s + 1/6·36-s + 0.328·37-s − 0.162·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.51879126582704, −13.17692394163426, −12.80375119987920, −12.12726350726775, −11.75871154482809, −11.08262369077091, −10.69179861610719, −10.27440978553905, −9.668723860389905, −9.272570907595551, −8.930004351372248, −8.219825923642584, −7.792926115188649, −7.455212565902597, −7.051015596168633, −6.237338964607226, −5.642254363018188, −5.421434834935375, −4.412255408015192, −3.948923541520199, −3.455337985672427, −2.501736588243722, −2.286893395872355, −1.720237303396679, −0.7183425979862435, 0,
0.7183425979862435, 1.720237303396679, 2.286893395872355, 2.501736588243722, 3.455337985672427, 3.948923541520199, 4.412255408015192, 5.421434834935375, 5.642254363018188, 6.237338964607226, 7.051015596168633, 7.455212565902597, 7.792926115188649, 8.219825923642584, 8.930004351372248, 9.272570907595551, 9.668723860389905, 10.27440978553905, 10.69179861610719, 11.08262369077091, 11.75871154482809, 12.12726350726775, 12.80375119987920, 13.17692394163426, 13.51879126582704