| L(s) = 1 | + i·3-s + (−2.09 − 0.793i)5-s + 4.15i·7-s − 9-s − 2.25·11-s + 0.730i·13-s + (0.793 − 2.09i)15-s + 4.88i·17-s − 4.91·19-s − 4.15·21-s − i·23-s + (3.73 + 3.31i)25-s − i·27-s + 7.68·29-s − 4.81·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.934 − 0.355i)5-s + 1.57i·7-s − 0.333·9-s − 0.680·11-s + 0.202i·13-s + (0.204 − 0.539i)15-s + 1.18i·17-s − 1.12·19-s − 0.907·21-s − 0.208i·23-s + (0.747 + 0.663i)25-s − 0.192i·27-s + 1.42·29-s − 0.864·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08461511357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08461511357\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.09 + 0.793i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 4.15iT - 7T^{2} \) |
| 11 | \( 1 + 2.25T + 11T^{2} \) |
| 13 | \( 1 - 0.730iT - 13T^{2} \) |
| 17 | \( 1 - 4.88iT - 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 + 4.81T + 31T^{2} \) |
| 37 | \( 1 - 1.42iT - 37T^{2} \) |
| 41 | \( 1 + 5.48T + 41T^{2} \) |
| 43 | \( 1 + 2.69iT - 43T^{2} \) |
| 47 | \( 1 + 13.4iT - 47T^{2} \) |
| 53 | \( 1 - 6.11iT - 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 2.27T + 61T^{2} \) |
| 67 | \( 1 + 7.22iT - 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 4.89T + 89T^{2} \) |
| 97 | \( 1 - 2.40iT - 97T^{2} \) |
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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
−9.027544355269674582480044577448, −8.511240158373597759353212279178, −8.281673713041210053137680207321, −7.07699498082697734572517908951, −6.12431087072124248375104391326, −5.39914140395998556591898797875, −4.66449699351458972326114396578, −3.84171015869613175091614829164, −2.88718584721222094303411649346, −1.92526456117574736334326557110,
0.03138493197987921642901282857, 1.02295682351700579134994857972, 2.54813157809469978261547564190, 3.40356035204937598726065094160, 4.30604680513130081269114500240, 4.97905292838398951609083581938, 6.26380265087675811949462099174, 7.02080694408773753985848514767, 7.44626049050098737303864620513, 8.065496156071128702579410985197
