L(s) = 1 | + 2i·7-s − 30·11-s + 4i·13-s − 90i·17-s + 28·19-s + 120i·23-s + 210·29-s − 4·31-s + 200i·37-s − 240·41-s + 136i·43-s + 120i·47-s + 339·49-s − 30i·53-s − 450·59-s + ⋯ |
L(s) = 1 | + 0.107i·7-s − 0.822·11-s + 0.0853i·13-s − 1.28i·17-s + 0.338·19-s + 1.08i·23-s + 1.34·29-s − 0.0231·31-s + 0.888i·37-s − 0.914·41-s + 0.482i·43-s + 0.372i·47-s + 0.988·49-s − 0.0777i·53-s − 0.992·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.548398105\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.548398105\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2iT - 343T^{2} \) |
| 11 | \( 1 + 30T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 90iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 28T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 210T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 200iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 240T + 6.89e4T^{2} \) |
| 43 | \( 1 - 136iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 120iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 30iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 450T + 2.05e5T^{2} \) |
| 61 | \( 1 + 166T + 2.26e5T^{2} \) |
| 67 | \( 1 - 908iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 250iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 916T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.14e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 420T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3iT - 9.12e5T^{2} \) |
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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
−9.859958741930050112163421641324, −9.108312963500059114962368417645, −8.118443000210158932411939371525, −7.39385929967941870811973318823, −6.48105692284846780380498347276, −5.36829310564883795972252311327, −4.72278427671724149084418650943, −3.35207164519829761499528186795, −2.45856659559515618628989173959, −0.984316828859506754651978103772,
0.47272824774670665840605765436, 1.94750423049145117134403054181, 3.08247278493990379848327026725, 4.20270811482181738308586973216, 5.17244807287499468475779863356, 6.12306024057428451509262079667, 6.99478329200163005348377374548, 8.045841514149026019984096396970, 8.560655716411356922320459271382, 9.669766547832549444602307137189