L(s) = 1 | − 3-s − 5-s + 3.24·7-s + 9-s + 1.35·11-s + 1.59·13-s + 15-s − 1.89·17-s − 6.84·19-s − 3.24·21-s + 4.02·23-s + 25-s − 27-s + 2.30·29-s − 31-s − 1.35·33-s − 3.24·35-s − 8.08·37-s − 1.59·39-s + 3.08·41-s − 7.92·43-s − 45-s − 5.38·47-s + 3.54·49-s + 1.89·51-s − 11.3·53-s − 1.35·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.22·7-s + 0.333·9-s + 0.408·11-s + 0.442·13-s + 0.258·15-s − 0.460·17-s − 1.56·19-s − 0.708·21-s + 0.839·23-s + 0.200·25-s − 0.192·27-s + 0.427·29-s − 0.179·31-s − 0.236·33-s − 0.548·35-s − 1.32·37-s − 0.255·39-s + 0.481·41-s − 1.20·43-s − 0.149·45-s − 0.785·47-s + 0.505·49-s + 0.265·51-s − 1.55·53-s − 0.182·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 - 1.59T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 23 | \( 1 - 4.02T + 23T^{2} \) |
| 29 | \( 1 - 2.30T + 29T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 - 3.08T + 41T^{2} \) |
| 43 | \( 1 + 7.92T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 - 0.514T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 5.43T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 2.67T + 83T^{2} \) |
| 89 | \( 1 - 6.83T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−7.56201711664589894266728121364, −6.71052898355355194010219389822, −6.33260971772502245357286252849, −5.26802772009764276509237412422, −4.72573724368274635489877631737, −4.15429038412858646290606787680, −3.24868793261115651049577020972, −2.00943571669209962936284411196, −1.31142455090167143081737890396, 0,
1.31142455090167143081737890396, 2.00943571669209962936284411196, 3.24868793261115651049577020972, 4.15429038412858646290606787680, 4.72573724368274635489877631737, 5.26802772009764276509237412422, 6.33260971772502245357286252849, 6.71052898355355194010219389822, 7.56201711664589894266728121364