| 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 | \( 1 + 0.0752iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.834957845753005768223021708037, −7.62425983913695498266442964288, −7.34810255966012572803503950827, −6.71245827555601111673022217739, −5.72405342890793037837251475854, −4.81708466972296392510363291904, −3.78759605215761507460754911828, −3.04618794883218011781178449537, −1.90289301336293635173673662840, −0.828497488124668619452077416165,
1.29261439554456744566595296938, 2.60986463461889397256333640603, 2.89499469014310709497560015470, 4.32195342639070495708564542700, 5.46807377846873291104481543366, 5.93999406205987939014262274113, 6.60204713615873523261608236811, 7.70236938014656988440274097301, 8.153081070690490060480829758712, 8.921793411111205892904507479155
