L(s) = 1 | − 3-s − 0.736·5-s + 1.91·7-s + 9-s + 3.35·11-s + 3.35·13-s + 0.736·15-s − 1.53·17-s − 8.41·19-s − 1.91·21-s − 1.97·23-s − 4.45·25-s − 27-s + 2.87·29-s + 2.41·31-s − 3.35·33-s − 1.40·35-s − 2.73·37-s − 3.35·39-s − 8.34·41-s + 1.91·43-s − 0.736·45-s + 3.72·47-s − 3.34·49-s + 1.53·51-s − 2.49·53-s − 2.46·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.329·5-s + 0.723·7-s + 0.333·9-s + 1.01·11-s + 0.931·13-s + 0.190·15-s − 0.372·17-s − 1.93·19-s − 0.417·21-s − 0.412·23-s − 0.891·25-s − 0.192·27-s + 0.533·29-s + 0.433·31-s − 0.583·33-s − 0.238·35-s − 0.450·37-s − 0.537·39-s − 1.30·41-s + 0.291·43-s − 0.109·45-s + 0.543·47-s − 0.477·49-s + 0.215·51-s − 0.342·53-s − 0.332·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 503 | \( 1 - T \) |
good | 5 | \( 1 + 0.736T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 - 3.35T + 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 + 8.41T + 19T^{2} \) |
| 23 | \( 1 + 1.97T + 23T^{2} \) |
| 29 | \( 1 - 2.87T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 8.34T + 41T^{2} \) |
| 43 | \( 1 - 1.91T + 43T^{2} \) |
| 47 | \( 1 - 3.72T + 47T^{2} \) |
| 53 | \( 1 + 2.49T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 0.175T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 - 5.41T + 71T^{2} \) |
| 73 | \( 1 - 1.32T + 73T^{2} \) |
| 79 | \( 1 - 4.15T + 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 + 4.80T + 89T^{2} \) |
| 97 | \( 1 + 7.00T + 97T^{2} \) |
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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
−7.84307763234508035640009583579, −6.77566398548732315326403946420, −6.37664206532421598445524177009, −5.70854914928583571314711060244, −4.60197385117413998034276850642, −4.25266982303126004384808852889, −3.43652555717961709809962465402, −2.06259165395853570015288621548, −1.35462665182713482784805941562, 0,
1.35462665182713482784805941562, 2.06259165395853570015288621548, 3.43652555717961709809962465402, 4.25266982303126004384808852889, 4.60197385117413998034276850642, 5.70854914928583571314711060244, 6.37664206532421598445524177009, 6.77566398548732315326403946420, 7.84307763234508035640009583579