L(s) = 1 | + 1.08·5-s − 6.01i·7-s + 17.7i·11-s − 12.4·13-s + 26.3·17-s + 5.19i·19-s − 29.8i·23-s − 23.8·25-s + 4.32·29-s + 44.8i·31-s − 6.49i·35-s + 20.4·37-s − 59.2·41-s + 19.1i·43-s + 41.0i·47-s + ⋯ |
L(s) = 1 | + 0.216·5-s − 0.859i·7-s + 1.61i·11-s − 0.955·13-s + 1.55·17-s + 0.273i·19-s − 1.29i·23-s − 0.953·25-s + 0.149·29-s + 1.44i·31-s − 0.185i·35-s + 0.551·37-s − 1.44·41-s + 0.445i·43-s + 0.873i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.397582446\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.397582446\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.08T + 25T^{2} \) |
| 7 | \( 1 + 6.01iT - 49T^{2} \) |
| 11 | \( 1 - 17.7iT - 121T^{2} \) |
| 13 | \( 1 + 12.4T + 169T^{2} \) |
| 17 | \( 1 - 26.3T + 289T^{2} \) |
| 19 | \( 1 - 5.19iT - 361T^{2} \) |
| 23 | \( 1 + 29.8iT - 529T^{2} \) |
| 29 | \( 1 - 4.32T + 841T^{2} \) |
| 31 | \( 1 - 44.8iT - 961T^{2} \) |
| 37 | \( 1 - 20.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 59.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 19.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 70.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 28.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 18.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 94.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 83.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 55.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 41.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 35.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 26.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 76.3T + 9.40e3T^{2} \) |
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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
−9.733529362428779816534260982082, −8.441854333943751231036713036598, −7.56863952011801451401802465417, −7.13610723305663624645258704799, −6.21522278217855173531178315881, −5.04710759592555398584652642483, −4.50718567939711620544387994834, −3.43245641244953114626264210812, −2.27297692732726088702685434292, −1.18730474776151180406231443087,
0.38417128212746178585976049721, 1.82250269592757573877161536328, 2.95542134380065904348034268392, 3.66802795776475342097279068223, 5.11210966968408532300626517714, 5.67181437883752390034700338727, 6.29574703767082463216722188807, 7.60289477364322778436817585976, 8.041958798148712134837542089427, 9.049916325025588334418921513236