L(s) = 1 | + 3-s − 3.54·7-s + 9-s − 2.20·11-s − 7.17·13-s + 6.36·17-s − 2.31·19-s − 3.54·21-s + 2.17·23-s + 27-s + 0.847·29-s − 4.30·31-s − 2.20·33-s − 7.22·37-s − 7.17·39-s − 1.34·41-s + 8.18·43-s + 6.05·47-s + 5.58·49-s + 6.36·51-s + 11.9·53-s − 2.31·57-s − 12.4·59-s − 6.92·61-s − 3.54·63-s + 4.73·67-s + 2.17·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34·7-s + 0.333·9-s − 0.665·11-s − 1.99·13-s + 1.54·17-s − 0.530·19-s − 0.774·21-s + 0.453·23-s + 0.192·27-s + 0.157·29-s − 0.772·31-s − 0.384·33-s − 1.18·37-s − 1.14·39-s − 0.209·41-s + 1.24·43-s + 0.883·47-s + 0.797·49-s + 0.891·51-s + 1.64·53-s − 0.306·57-s − 1.62·59-s − 0.886·61-s − 0.446·63-s + 0.578·67-s + 0.261·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.376488613\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.376488613\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 + 7.17T + 13T^{2} \) |
| 17 | \( 1 - 6.36T + 17T^{2} \) |
| 19 | \( 1 + 2.31T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 - 0.847T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + 7.22T + 37T^{2} \) |
| 41 | \( 1 + 1.34T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 - 6.05T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 4.73T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 1.08T + 73T^{2} \) |
| 79 | \( 1 + 5.84T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 7.02T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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
−7.67759083378433894797728806128, −7.33160012905300564335476781207, −6.68898379065394421887708229284, −5.68886416301788425498848950804, −5.17160158440300194831045736917, −4.22388139198733691553844006660, −3.31760596015894463646989711883, −2.81397985564879230601524384515, −2.04613697108456957996472743505, −0.53857581947606624304283391254,
0.53857581947606624304283391254, 2.04613697108456957996472743505, 2.81397985564879230601524384515, 3.31760596015894463646989711883, 4.22388139198733691553844006660, 5.17160158440300194831045736917, 5.68886416301788425498848950804, 6.68898379065394421887708229284, 7.33160012905300564335476781207, 7.67759083378433894797728806128