L(s) = 1 | + 2.14i·3-s − 1.14i·7-s − 1.60·9-s + 5.89·11-s + 4.89i·13-s − 5.89i·17-s + 2.34·19-s + 2.45·21-s − i·23-s + 3.00i·27-s − 3.74·29-s + 5.68·31-s + 12.6i·33-s + 4i·37-s − 10.4·39-s + ⋯ |
L(s) = 1 | + 1.23i·3-s − 0.432i·7-s − 0.533·9-s + 1.77·11-s + 1.35i·13-s − 1.42i·17-s + 0.538·19-s + 0.536·21-s − 0.208i·23-s + 0.577i·27-s − 0.695·29-s + 1.02·31-s + 2.20i·33-s + 0.657i·37-s − 1.68·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.353869447\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.353869447\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 - 2.14iT - 3T^{2} \) |
| 7 | \( 1 + 1.14iT - 7T^{2} \) |
| 11 | \( 1 - 5.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 5.89iT - 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 7.74iT - 47T^{2} \) |
| 53 | \( 1 + 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 0.797T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 15.5iT - 67T^{2} \) |
| 71 | \( 1 - 2.94T + 71T^{2} \) |
| 73 | \( 1 + 6.32iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 0.912iT - 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
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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
−8.876991173894774500159533445813, −7.67001691405298536189260137580, −6.81778489746481608507098660483, −6.47229399241828668527458561107, −5.19027630655108844096440801191, −4.66656246090835631506269127345, −3.88581515319703185783351045167, −3.49234594401900469184531078792, −2.13256248045723337378687682754, −0.937625008117575699146003111847,
0.907261291625832956941011649745, 1.55019230782408472121306907292, 2.56279500520851584166569270445, 3.56994557156115374059874746751, 4.33333153287288943236617017286, 5.68450864140631065343569187143, 5.98593536459781710849982527412, 6.80555748925419487343258283847, 7.39041522636624481299074977858, 8.214695765612354827721051065873