L(s) = 1 | + (−1.45 + 1.45i)2-s + 1.81i·3-s − 2.21i·4-s + (2.01 + 2.01i)5-s + (−2.64 − 2.64i)6-s + (−2.38 − 1.13i)7-s + (0.311 + 0.311i)8-s − 0.311·9-s − 5.84·10-s + (0.451 + 0.451i)11-s + 4.02·12-s + (−3.40 − 1.19i)13-s + (5.11 − 1.81i)14-s + (−3.66 + 3.66i)15-s + 3.52·16-s + 4.32·17-s + ⋯ |
L(s) = 1 | + (−1.02 + 1.02i)2-s + 1.05i·3-s − 1.10i·4-s + (0.900 + 0.900i)5-s + (−1.07 − 1.07i)6-s + (−0.903 − 0.429i)7-s + (0.109 + 0.109i)8-s − 0.103·9-s − 1.84·10-s + (0.136 + 0.136i)11-s + 1.16·12-s + (−0.943 − 0.330i)13-s + (1.36 − 0.486i)14-s + (−0.946 + 0.946i)15-s + 0.881·16-s + 1.04·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.151978 + 0.611802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.151978 + 0.611802i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.38 + 1.13i)T \) |
| 13 | \( 1 + (3.40 + 1.19i)T \) |
good | 2 | \( 1 + (1.45 - 1.45i)T - 2iT^{2} \) |
| 3 | \( 1 - 1.81iT - 3T^{2} \) |
| 5 | \( 1 + (-2.01 - 2.01i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.451 - 0.451i)T + 11iT^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + (-3.40 - 3.40i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.933iT - 23T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 + (5.47 + 5.47i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.14 - 2.14i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.81 + 1.81i)T + 41iT^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + (5.90 - 5.90i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.36T + 53T^{2} \) |
| 59 | \( 1 + (-0.255 + 0.255i)T - 59iT^{2} \) |
| 61 | \( 1 + 7.78iT - 61T^{2} \) |
| 67 | \( 1 + (-7.28 + 7.28i)T - 67iT^{2} \) |
| 71 | \( 1 + (-5.56 + 5.56i)T - 71iT^{2} \) |
| 73 | \( 1 + (-8.86 + 8.86i)T - 73iT^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + (4.30 + 4.30i)T + 83iT^{2} \) |
| 89 | \( 1 + (5.61 - 5.61i)T - 89iT^{2} \) |
| 97 | \( 1 + (-0.236 - 0.236i)T + 97iT^{2} \) |
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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
−14.84454404594236574706348348025, −14.02307424099426455281774538619, −12.45075795232940762538743852721, −10.48266970126546985505565146700, −9.870028258384748866932033611954, −9.485346532910620107157420710337, −7.69067557291957051376456478089, −6.69507288799705621013217993158, −5.53075694838033038587537580494, −3.39253255635364349815800858537,
1.21962236231779655836807211588, 2.67919913603457126081096348929, 5.47043697906506938981431486186, 6.95089972379364751382851545317, 8.403962099274058684554633377065, 9.487310415668599938723906840695, 9.986992120056384237455588432038, 11.71819546929242356182358221774, 12.49821278450096654620137069371, 13.08820761844759144469368410657