L(s) = 1 | + 1.41i·2-s − 3-s − 1.00·4-s − 1.41i·6-s + 1.41i·7-s + 11-s + 1.00·12-s − 13-s − 2.00·14-s − 0.999·16-s − 17-s − 1.41i·19-s − 1.41i·21-s + 1.41i·22-s + 1.41i·23-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 3-s − 1.00·4-s − 1.41i·6-s + 1.41i·7-s + 11-s + 1.00·12-s − 13-s − 2.00·14-s − 0.999·16-s − 17-s − 1.41i·19-s − 1.41i·21-s + 1.41i·22-s + 1.41i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 491 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 491 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5474355833\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5474355833\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 491 | \( 1 + T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−11.65719507675183248321390873326, −11.01394189681575340618491921185, −9.234197510472575963424399304142, −9.082971528398577265618048125968, −7.77714775459169516128406685163, −6.77614374531070989209157455430, −6.12016079536639950028263187832, −5.36174808436956649741684526432, −4.61061910095039302457820578432, −2.50080864522921977867982890070,
0.78529984351718831012220480159, 2.39184439540814798476143127275, 4.04660703062382259730405063948, 4.43240544172426865917037987903, 6.10477602340803744055610761972, 6.86914451524692462209677981684, 8.105090678403379605310251124499, 9.556299453522206533871549070297, 10.12121881750443426407307071197, 10.88720377732220601393531824489