| L(s) = 1 | + 3-s − 1.69·5-s + 2.31·7-s + 9-s + 0.271·11-s + 0.381·13-s − 1.69·15-s − 3.91·17-s − 7.86·19-s + 2.31·21-s + 6.38·23-s − 2.14·25-s + 27-s + 1.34·29-s − 3.51·31-s + 0.271·33-s − 3.91·35-s − 3.32·37-s + 0.381·39-s + 4.64·41-s + 3.26·43-s − 1.69·45-s − 8.46·47-s − 1.63·49-s − 3.91·51-s + 0.738·53-s − 0.459·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.756·5-s + 0.875·7-s + 0.333·9-s + 0.0819·11-s + 0.105·13-s − 0.436·15-s − 0.949·17-s − 1.80·19-s + 0.505·21-s + 1.33·23-s − 0.428·25-s + 0.192·27-s + 0.250·29-s − 0.631·31-s + 0.0472·33-s − 0.662·35-s − 0.547·37-s + 0.0610·39-s + 0.725·41-s + 0.497·43-s − 0.252·45-s − 1.23·47-s − 0.232·49-s − 0.548·51-s + 0.101·53-s − 0.0619·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 11 | \( 1 - 0.271T + 11T^{2} \) |
| 13 | \( 1 - 0.381T + 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 + 7.86T + 19T^{2} \) |
| 23 | \( 1 - 6.38T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 + 3.32T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 + 8.46T + 47T^{2} \) |
| 53 | \( 1 - 0.738T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 - 4.19T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 0.142T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
| 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
−7.86225062051063443322431465961, −7.09079653005822063453059796377, −6.51925984839868000111873009680, −5.48662299314970453759011267850, −4.47561960803358511126038407736, −4.25575312959605850346349192255, −3.23871419213197579553752686175, −2.30571518139459028506592888802, −1.47746816552786885456304699978, 0,
1.47746816552786885456304699978, 2.30571518139459028506592888802, 3.23871419213197579553752686175, 4.25575312959605850346349192255, 4.47561960803358511126038407736, 5.48662299314970453759011267850, 6.51925984839868000111873009680, 7.09079653005822063453059796377, 7.86225062051063443322431465961
