L(s) = 1 | − 0.933·5-s + (−2.45 − 0.981i)7-s + 1.75i·11-s + 4.20i·13-s + 0.231·17-s − 5.00i·19-s − 2.61i·23-s − 4.12·25-s − 3.77i·29-s − 0.840i·31-s + (2.29 + 0.915i)35-s + 3.01·37-s + 8.98·41-s − 2.40·43-s + 1.03·47-s + ⋯ |
L(s) = 1 | − 0.417·5-s + (−0.928 − 0.371i)7-s + 0.529i·11-s + 1.16i·13-s + 0.0560·17-s − 1.14i·19-s − 0.545i·23-s − 0.825·25-s − 0.700i·29-s − 0.150i·31-s + (0.387 + 0.154i)35-s + 0.495·37-s + 1.40·41-s − 0.366·43-s + 0.150·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.224458223\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.224458223\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.45 + 0.981i)T \) |
good | 5 | \( 1 + 0.933T + 5T^{2} \) |
| 11 | \( 1 - 1.75iT - 11T^{2} \) |
| 13 | \( 1 - 4.20iT - 13T^{2} \) |
| 17 | \( 1 - 0.231T + 17T^{2} \) |
| 19 | \( 1 + 5.00iT - 19T^{2} \) |
| 23 | \( 1 + 2.61iT - 23T^{2} \) |
| 29 | \( 1 + 3.77iT - 29T^{2} \) |
| 31 | \( 1 + 0.840iT - 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 - 8.98T + 41T^{2} \) |
| 43 | \( 1 + 2.40T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 - 9.04iT - 53T^{2} \) |
| 59 | \( 1 - 6.77T + 59T^{2} \) |
| 61 | \( 1 - 8.93iT - 61T^{2} \) |
| 67 | \( 1 + 1.10T + 67T^{2} \) |
| 71 | \( 1 + 7.21iT - 71T^{2} \) |
| 73 | \( 1 - 3.44iT - 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 1.63iT - 97T^{2} \) |
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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
−8.840835763970725298160708987993, −7.76479778679911378712428561116, −7.16533704291169369067847764340, −6.51795050879697183098578794686, −5.76811148457698631441290942021, −4.43349138546754163781532802373, −4.18850977898720852494340523486, −3.01693247426164675270512853083, −2.10452651003610476733163997784, −0.57921180230010721327491015717,
0.74411759913879304394623141433, 2.24654137567586051453626169011, 3.38929885864248993989306252575, 3.68613113591769369611489127076, 5.04909884154934673196488343104, 5.81906397713802163100491728523, 6.33267142810182078058007138878, 7.42778677788262279962702782714, 7.981460544069878736482291268248, 8.708563317535161465228892072384