L(s) = 1 | + (0.5 − 2.78i)2-s + (−7.49 − 2.78i)4-s − 2·5-s + 33.4i·7-s + (−11.4 + 19.4i)8-s − 27·9-s + (−1 + 5.56i)10-s + (93 + 16.7i)14-s + (48.4 + 41.7i)16-s + (−13.5 + 75.1i)18-s + 55.6i·19-s + (14.9 + 5.56i)20-s − 121·25-s + (93 − 250. i)28-s + 172. i·31-s + (140. − 114. i)32-s + ⋯ |
L(s) = 1 | + (0.176 − 0.984i)2-s + (−0.937 − 0.347i)4-s − 0.178·5-s + 1.80i·7-s + (−0.508 + 0.861i)8-s − 9-s + (−0.0316 + 0.176i)10-s + (1.77 + 0.318i)14-s + (0.757 + 0.652i)16-s + (−0.176 + 0.984i)18-s + 0.672i·19-s + (0.167 + 0.0622i)20-s − 0.967·25-s + (0.627 − 1.69i)28-s + 0.999i·31-s + (0.776 − 0.630i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.620947 + 0.431857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.620947 + 0.431857i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 2.78i)T \) |
| 31 | \( 1 - 172. iT \) |
good | 3 | \( 1 + 27T^{2} \) |
| 5 | \( 1 + 2T + 125T^{2} \) |
| 7 | \( 1 - 33.4iT - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 - 55.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 278T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 189. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 + 523. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 857. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 656. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.90e3T + 9.12e5T^{2} \) |
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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
−12.81268344637674264412799311812, −11.92391377500608265827532219908, −11.45203557702265082189986049585, −10.05350103809888349052676677392, −8.923107405462924579964981770043, −8.291986757122217922277701682317, −6.00192615884594482724549271996, −5.16300131950600311541676092085, −3.33298338377701237097363509113, −2.12503024523908125777662830582,
0.36425813691943199516077323340, 3.54136928556152791095773913448, 4.69108481642176139809910298238, 6.15548640948323978985853591114, 7.29244739467627077681133455726, 8.098104056701677226481838952610, 9.394121043452456148714338570571, 10.58056503746539082675882648508, 11.74354484781918643953353949918, 13.24965469543222111953890737899