L(s) = 1 | − 3.31i·5-s − i·7-s − 4.72·11-s + 4.97·13-s + 0.484i·17-s + 2.29i·19-s − 7.97·23-s − 5.97·25-s − 1.41i·29-s − 7.66i·31-s − 3.31·35-s + 2.39·37-s + 6.55i·41-s − 5.37i·43-s − 6.21·47-s + ⋯ |
L(s) = 1 | − 1.48i·5-s − 0.377i·7-s − 1.42·11-s + 1.38·13-s + 0.117i·17-s + 0.525i·19-s − 1.66·23-s − 1.19·25-s − 0.262i·29-s − 1.37i·31-s − 0.560·35-s + 0.393·37-s + 1.02i·41-s − 0.819i·43-s − 0.906·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3954219695\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3954219695\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 3.31iT - 5T^{2} \) |
| 11 | \( 1 + 4.72T + 11T^{2} \) |
| 13 | \( 1 - 4.97T + 13T^{2} \) |
| 17 | \( 1 - 0.484iT - 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 + 7.97T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 7.66iT - 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 - 6.55iT - 41T^{2} \) |
| 43 | \( 1 + 5.37iT - 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 1.00iT - 53T^{2} \) |
| 59 | \( 1 - 1.38T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 3.27iT - 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 + 2.10T + 73T^{2} \) |
| 79 | \( 1 + 12.0iT - 79T^{2} \) |
| 83 | \( 1 + 3.24T + 83T^{2} \) |
| 89 | \( 1 - 5.72iT - 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.946248228624446998836589217979, −7.73379031910990556241205801914, −6.20965286112737230381058334422, −5.84465115877036228946681210502, −4.94166135517582561639293686882, −4.26454159560204237363616811196, −3.51276252754482944860719173840, −2.19296232636556929616015510186, −1.23606982104351875331330739030, −0.11235887295961448298586996020,
1.74357723956476554793062401648, 2.76138026143399388495547430594, 3.23439284884824747630822249099, 4.23409663997593550323662894751, 5.33772035957502628886164385392, 6.02041743074750139597843706063, 6.63620241855020103985671122527, 7.41068811804609919593663050297, 8.107299099331134405279938160259, 8.715202757262024828889148138560