L(s) = 1 | + i·2-s − 4-s + 1.84·5-s − i·8-s + 1.84i·10-s + 0.765i·13-s + 16-s − 0.765·17-s − 1.84·20-s + 2.41·25-s − 0.765·26-s + i·32-s − 0.765i·34-s + 1.41·37-s − 1.84i·40-s − 0.765·41-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s + 1.84·5-s − i·8-s + 1.84i·10-s + 0.765i·13-s + 16-s − 0.765·17-s − 1.84·20-s + 2.41·25-s − 0.765·26-s + i·32-s − 0.765i·34-s + 1.41·37-s − 1.84i·40-s − 0.765·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.387183319\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.387183319\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.84T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 0.765iT - T^{2} \) |
| 17 | \( 1 + 0.765T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 + 0.765T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.84iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.84iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.84T + T^{2} \) |
| 97 | \( 1 - 1.84iT - T^{2} \) |
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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
−9.449228039087642369900110934950, −8.995531086374524324565891613718, −8.142709213880847615094332474154, −6.95113624248823697267981143064, −6.49458202630108108758718598442, −5.75842209397954328063701618222, −5.03879713414506283038004542888, −4.14767984974902007584685434526, −2.67107320136561336442808555938, −1.52861518672177461537377006355,
1.28862775764033900154475398541, 2.30140401196221761589119658538, 2.98548705900688903005765360359, 4.32682505765461180980622422072, 5.26079863255650564321353082630, 5.85524836894980512040236051750, 6.74090158127873025714112256832, 8.040133557904187252417891712685, 8.860008671871103150296670583598, 9.550886210795606231658646905120