L(s) = 1 | + 2.82·5-s + (−2.44 − i)7-s − 3.46·11-s + 3.00·25-s + 10.3i·29-s − 4.89·31-s + (−6.92 − 2.82i)35-s + (4.99 + 4.89i)49-s − 3.46i·53-s − 9.79·55-s + 11.3i·59-s + 9.79i·73-s + (8.48 + 3.46i)77-s + 10i·79-s + 5.65i·83-s + ⋯ |
L(s) = 1 | + 1.26·5-s + (−0.925 − 0.377i)7-s − 1.04·11-s + 0.600·25-s + 1.92i·29-s − 0.879·31-s + (−1.17 − 0.478i)35-s + (0.714 + 0.699i)49-s − 0.475i·53-s − 1.32·55-s + 1.47i·59-s + 1.14i·73-s + (0.966 + 0.394i)77-s + 1.12i·79-s + 0.620i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9821192736\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9821192736\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.44 + i)T \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 19.5iT - 97T^{2} \) |
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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
−8.871531114061864561054446165184, −7.899353630804204623764740711648, −7.03557350670993834446207969602, −6.53825627977569740862803984495, −5.53734432438082803590974220635, −5.29040501007214750722301399409, −4.04339650295518871358367145548, −3.08247758275837325953241125170, −2.37061393421278374202692685318, −1.27047072984039802833428090041,
0.26342065847264625188858095395, 1.88592290276557561194805912986, 2.54292078117171619698970002659, 3.37769650134025672643571444519, 4.52674197733062831828512927110, 5.49911152781506377693351479853, 5.90846157167292771871969483487, 6.56331847495906753285641611397, 7.47337037474231566114014700896, 8.239848272653087263335536393624