L(s) = 1 | + 0.193i·2-s + 1.96·4-s + 3.35i·7-s + 0.768i·8-s − 11-s − 2.96i·13-s − 0.649·14-s + 3.77·16-s + 4.57i·17-s + 4.31·19-s − 0.193i·22-s − 6.70i·23-s + 0.574·26-s + 6.57i·28-s − 3.61·29-s + ⋯ |
L(s) = 1 | + 0.137i·2-s + 0.981·4-s + 1.26i·7-s + 0.271i·8-s − 0.301·11-s − 0.821i·13-s − 0.173·14-s + 0.943·16-s + 1.10i·17-s + 0.989·19-s − 0.0413i·22-s − 1.39i·23-s + 0.112·26-s + 1.24i·28-s − 0.670·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.349066840\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.349066840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 0.193iT - 2T^{2} \) |
| 7 | \( 1 - 3.35iT - 7T^{2} \) |
| 13 | \( 1 + 2.96iT - 13T^{2} \) |
| 17 | \( 1 - 4.57iT - 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 + 6.70iT - 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 4.38T + 41T^{2} \) |
| 43 | \( 1 - 9.27iT - 43T^{2} \) |
| 47 | \( 1 - 9.92iT - 47T^{2} \) |
| 53 | \( 1 - 4.70iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 - 5.92iT - 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 7.73iT - 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 2.77T + 89T^{2} \) |
| 97 | \( 1 - 0.0752iT - 97T^{2} \) |
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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
−8.921793411111205892904507479155, −8.153081070690490060480829758712, −7.70236938014656988440274097301, −6.60204713615873523261608236811, −5.93999406205987939014262274113, −5.46807377846873291104481543366, −4.32195342639070495708564542700, −2.89499469014310709497560015470, −2.60986463461889397256333640603, −1.29261439554456744566595296938,
0.828497488124668619452077416165, 1.90289301336293635173673662840, 3.04618794883218011781178449537, 3.78759605215761507460754911828, 4.81708466972296392510363291904, 5.72405342890793037837251475854, 6.71245827555601111673022217739, 7.34810255966012572803503950827, 7.62425983913695498266442964288, 8.834957845753005768223021708037