L(s) = 1 | − 1.50i·3-s + 4.31i·5-s − 4.93i·7-s + 0.739·9-s + 1.58i·11-s − 0.143·13-s + 6.48·15-s + (−0.180 + 4.11i)17-s + 0.296·19-s − 7.41·21-s + 6.24i·23-s − 13.5·25-s − 5.62i·27-s + 6.17i·29-s − 4.07i·31-s + ⋯ |
L(s) = 1 | − 0.868i·3-s + 1.92i·5-s − 1.86i·7-s + 0.246·9-s + 0.477i·11-s − 0.0398·13-s + 1.67·15-s + (−0.0437 + 0.999i)17-s + 0.0680·19-s − 1.61·21-s + 1.30i·23-s − 2.71·25-s − 1.08i·27-s + 1.14i·29-s − 0.732i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0437 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0437 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.277511486\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277511486\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (0.180 - 4.11i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 1.50iT - 3T^{2} \) |
| 5 | \( 1 - 4.31iT - 5T^{2} \) |
| 7 | \( 1 + 4.93iT - 7T^{2} \) |
| 11 | \( 1 - 1.58iT - 11T^{2} \) |
| 13 | \( 1 + 0.143T + 13T^{2} \) |
| 19 | \( 1 - 0.296T + 19T^{2} \) |
| 23 | \( 1 - 6.24iT - 23T^{2} \) |
| 29 | \( 1 - 6.17iT - 29T^{2} \) |
| 31 | \( 1 + 4.07iT - 31T^{2} \) |
| 37 | \( 1 - 7.76iT - 37T^{2} \) |
| 41 | \( 1 - 1.59iT - 41T^{2} \) |
| 43 | \( 1 + 6.67T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 6.38T + 53T^{2} \) |
| 61 | \( 1 + 0.241iT - 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 1.01iT - 71T^{2} \) |
| 73 | \( 1 - 0.169iT - 73T^{2} \) |
| 79 | \( 1 + 11.2iT - 79T^{2} \) |
| 83 | \( 1 + 5.97T + 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 - 4.04iT - 97T^{2} \) |
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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
−8.137589395481108211903249687535, −7.64235644991130226311626890550, −7.06714153349816038518976871579, −6.73269704366558591089541230316, −6.12010575366843448295823570024, −4.74799910308078436617333941085, −3.70718391391548743028570150418, −3.37094443785990029765741152057, −2.06858053746216774467538465848, −1.29095597674915153552697941413,
0.36727481691341190935711618698, 1.72814496264692897592035193310, 2.68696742513477381582811300675, 3.86497866748564914982294613503, 4.70002737406065498731703008568, 5.19494718231353881573551798133, 5.64555345471794419153406303055, 6.62013965691821426643833391821, 7.932840476015882697000283683377, 8.625057886685036709836474774525