L(s) = 1 | − 1.41·2-s + (0.707 + 1.58i)3-s + 2.00·4-s + (−1.00 − 2.23i)6-s + 3.16i·7-s − 2.82·8-s + (−2.00 + 2.23i)9-s + (1.41 + 3.16i)12-s − 4.47i·14-s + 4.00·16-s + (2.82 − 3.16i)18-s + (−5.00 + 2.23i)21-s − 1.41·23-s + (−2.00 − 4.47i)24-s + (−4.94 − 1.58i)27-s + 6.32i·28-s + ⋯ |
L(s) = 1 | − 1.00·2-s + (0.408 + 0.912i)3-s + 1.00·4-s + (−0.408 − 0.912i)6-s + 1.19i·7-s − 1.00·8-s + (−0.666 + 0.745i)9-s + (0.408 + 0.912i)12-s − 1.19i·14-s + 1.00·16-s + (0.666 − 0.745i)18-s + (−1.09 + 0.487i)21-s − 0.294·23-s + (−0.408 − 0.912i)24-s + (−0.952 − 0.304i)27-s + 1.19i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.464894 + 0.717173i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.464894 + 0.717173i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (-0.707 - 1.58i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.16iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.94iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 3.16iT - 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.75675778843100714566471203762, −10.86487665919189777262031032586, −10.02909412493151551553869249102, −9.060520753114980386466674421723, −8.651298742971363718079070587365, −7.54662643931072457699598369477, −6.13738944215972820559193889449, −5.11244926398844520313607032892, −3.36592430985495431315241637102, −2.20948265216510993730510528330,
0.826896808199285001952706185328, 2.34879477507242529031403756799, 3.83745552114054671214763759206, 5.92609784288548712531003643413, 6.94745777030392306371897200553, 7.62346796558641939159658080491, 8.430185419055942607208072514309, 9.507040536944128564755309985589, 10.40023164374517050993345475509, 11.37708181572973165851678757763