| L(s) = 1 | − 0.198i·3-s − 1.31i·5-s + (0.640 + 2.56i)7-s + 2.96·9-s + (1.56 − 2.92i)11-s + 13-s − 0.260·15-s + 4.67·17-s + 4.13·19-s + (0.509 − 0.127i)21-s + 0.678·23-s + 3.28·25-s − 1.18i·27-s + 4.73i·29-s − 3.43i·31-s + ⋯ |
| L(s) = 1 | − 0.114i·3-s − 0.586i·5-s + (0.241 + 0.970i)7-s + 0.986·9-s + (0.471 − 0.881i)11-s + 0.277·13-s − 0.0672·15-s + 1.13·17-s + 0.948·19-s + (0.111 − 0.0277i)21-s + 0.141·23-s + 0.656·25-s − 0.227i·27-s + 0.879i·29-s − 0.616i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.501734227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.501734227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.640 - 2.56i)T \) |
| 11 | \( 1 + (-1.56 + 2.92i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.198iT - 3T^{2} \) |
| 5 | \( 1 + 1.31iT - 5T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 23 | \( 1 - 0.678T + 23T^{2} \) |
| 29 | \( 1 - 4.73iT - 29T^{2} \) |
| 31 | \( 1 + 3.43iT - 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 10.0iT - 43T^{2} \) |
| 47 | \( 1 + 2.08iT - 47T^{2} \) |
| 53 | \( 1 + 8.71T + 53T^{2} \) |
| 59 | \( 1 + 7.26iT - 59T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 9.55T + 73T^{2} \) |
| 79 | \( 1 - 1.30iT - 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 9.83iT - 89T^{2} \) |
| 97 | \( 1 - 14.3iT - 97T^{2} \) |
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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
−8.257782134073826226504805589216, −7.967913716223093962389585418478, −6.87117792181049347647583618446, −6.22356610997092343079991227144, −5.26535478538792937051769797885, −4.92703657955373590965008830084, −3.68651366810430247300931839790, −3.04738261063147083148300160155, −1.67953372098879180311861234810, −0.993787056238800455146089753081,
1.01057672028423471435588609733, 1.83406510688283171887211131331, 3.23235217299940557655224171478, 3.81056196475878513209991662020, 4.66592242779856258335625537120, 5.35649444945725626907925871931, 6.59444696669876240155020179630, 7.02122685912291357098994835888, 7.57025788533365872373983441986, 8.319242311702162094017682198947
