L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3·7-s − 8-s + 9-s − 3·11-s − 12-s − 13-s + 3·14-s + 16-s − 3·17-s − 18-s + 3·21-s + 3·22-s − 4·23-s + 24-s + 26-s − 27-s − 3·28-s + 5·29-s − 3·31-s − 32-s + 3·33-s + 3·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.654·21-s + 0.639·22-s − 0.834·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.566·28-s + 0.928·29-s − 0.538·31-s − 0.176·32-s + 0.522·33-s + 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5213108239\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5213108239\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.442658688949202449990693179407, −8.356779248398044699510513150888, −7.68508190509758357677163019337, −6.72735263754447773611649684611, −6.24918379406169666831171010852, −5.33317551635795333569993671797, −4.28628914641117018664838832502, −3.09685022220753397296375934858, −2.16705093681650178204775437434, −0.52377673393303732878330164226,
0.52377673393303732878330164226, 2.16705093681650178204775437434, 3.09685022220753397296375934858, 4.28628914641117018664838832502, 5.33317551635795333569993671797, 6.24918379406169666831171010852, 6.72735263754447773611649684611, 7.68508190509758357677163019337, 8.356779248398044699510513150888, 9.442658688949202449990693179407