L(s) = 1 | + 2.63i·2-s − 4.93·4-s + 1.79i·7-s − 7.73i·8-s − 1.80·11-s + 1.97i·13-s − 4.73·14-s + 10.4·16-s + 4.80i·17-s − 2.96·19-s − 4.76i·22-s + 1.73i·23-s − 5.19·26-s − 8.87i·28-s − 7.36·29-s + ⋯ |
L(s) = 1 | + 1.86i·2-s − 2.46·4-s + 0.679i·7-s − 2.73i·8-s − 0.545·11-s + 0.546i·13-s − 1.26·14-s + 2.62·16-s + 1.16i·17-s − 0.680·19-s − 1.01i·22-s + 0.361i·23-s − 1.01·26-s − 1.67i·28-s − 1.36·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1286962553\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1286962553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2.63iT - 2T^{2} \) |
| 7 | \( 1 - 1.79iT - 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 - 1.97iT - 13T^{2} \) |
| 17 | \( 1 - 4.80iT - 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 - 1.73iT - 23T^{2} \) |
| 29 | \( 1 + 7.36T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 11.6iT - 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 + 7.27iT - 43T^{2} \) |
| 47 | \( 1 - 6.29iT - 47T^{2} \) |
| 53 | \( 1 + 1.72iT - 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 9.10iT - 67T^{2} \) |
| 71 | \( 1 + 1.27T + 71T^{2} \) |
| 73 | \( 1 + 3.58iT - 73T^{2} \) |
| 79 | \( 1 + 2.11T + 79T^{2} \) |
| 83 | \( 1 + 1.09iT - 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 3.83iT - 97T^{2} \) |
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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
−9.251625612575961779024110142888, −8.989805789867674629974541203716, −8.122153165994000715759279685485, −7.49870288663812486240680639120, −6.75551566567580743396198809106, −5.75356517524663277794698269878, −5.61275491009330059431903629100, −4.42738794980325306021169536841, −3.70937545588801546141846898972, −2.03697679636929376988281790137,
0.05008132385277651810879759338, 1.19249968965481063077588709557, 2.40424900581142766609834776623, 3.13845017574912623029005258546, 4.06771497207481265903122280238, 4.80618441121291627450234316163, 5.63199177872783162995840862941, 7.01033845254934551274204331612, 7.936857404543978751125268533448, 8.699201695933586183179024414151