L(s) = 1 | + (0.270 − 0.270i)5-s + 7-s + (−3.03 − 3.03i)11-s + (1.28 − 1.28i)13-s − 5.15i·17-s + (5.55 + 5.55i)19-s + 5.69i·23-s + 4.85i·25-s + (1.94 + 1.94i)29-s + 0.936i·31-s + (0.270 − 0.270i)35-s + (−1.52 − 1.52i)37-s + 12.4·41-s + (0.346 − 0.346i)43-s − 9.71·47-s + ⋯ |
L(s) = 1 | + (0.120 − 0.120i)5-s + 0.377·7-s + (−0.915 − 0.915i)11-s + (0.357 − 0.357i)13-s − 1.25i·17-s + (1.27 + 1.27i)19-s + 1.18i·23-s + 0.970i·25-s + (0.360 + 0.360i)29-s + 0.168i·31-s + (0.0456 − 0.0456i)35-s + (−0.251 − 0.251i)37-s + 1.94·41-s + (0.0528 − 0.0528i)43-s − 1.41·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.972408847\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.972408847\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-0.270 + 0.270i)T - 5iT^{2} \) |
| 11 | \( 1 + (3.03 + 3.03i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.28 + 1.28i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.15iT - 17T^{2} \) |
| 19 | \( 1 + (-5.55 - 5.55i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.69iT - 23T^{2} \) |
| 29 | \( 1 + (-1.94 - 1.94i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.936iT - 31T^{2} \) |
| 37 | \( 1 + (1.52 + 1.52i)T + 37iT^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + (-0.346 + 0.346i)T - 43iT^{2} \) |
| 47 | \( 1 + 9.71T + 47T^{2} \) |
| 53 | \( 1 + (-7.69 + 7.69i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.03 - 2.03i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.86 + 5.86i)T - 61iT^{2} \) |
| 67 | \( 1 + (10.7 + 10.7i)T + 67iT^{2} \) |
| 71 | \( 1 + 12.5iT - 71T^{2} \) |
| 73 | \( 1 + 7.37iT - 73T^{2} \) |
| 79 | \( 1 - 13.0iT - 79T^{2} \) |
| 83 | \( 1 + (-2.53 + 2.53i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.04T + 89T^{2} \) |
| 97 | \( 1 - 0.334T + 97T^{2} \) |
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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
−8.212432760860196313979884387829, −7.71766441990756061178435079890, −7.10492316726283109302112064127, −5.85285194777610569278496105125, −5.49098080278026910750864957541, −4.80667438948513336221256683751, −3.48052061175059742580695038703, −3.08807499676065435218477238273, −1.77835042796090096585140609253, −0.71729316490474443573847560257,
0.945858504381566398775256464466, 2.21567600095674624077849192797, 2.81902994916872836232600987244, 4.15439856697911968151393467925, 4.64325800549198229970824086143, 5.54374412224323534804715458454, 6.32322954202210467168533942317, 7.11131792466953908633758718234, 7.76681802420280578127608239029, 8.500568630155362062395255641110