| L(s) = 1 | + 3-s − 4.01·5-s − 3.41·7-s + 9-s − 0.942·11-s − 0.0524·13-s − 4.01·15-s + 2.46·17-s + 6.45·19-s − 3.41·21-s − 1.91·23-s + 11.1·25-s + 27-s − 1.42·29-s + 3.96·31-s − 0.942·33-s + 13.6·35-s + 6.45·37-s − 0.0524·39-s + 4.79·41-s + 3.95·43-s − 4.01·45-s − 7.81·47-s + 4.63·49-s + 2.46·51-s − 5.37·53-s + 3.78·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.79·5-s − 1.28·7-s + 0.333·9-s − 0.284·11-s − 0.0145·13-s − 1.03·15-s + 0.598·17-s + 1.48·19-s − 0.744·21-s − 0.399·23-s + 2.22·25-s + 0.192·27-s − 0.264·29-s + 0.711·31-s − 0.164·33-s + 2.31·35-s + 1.06·37-s − 0.00840·39-s + 0.748·41-s + 0.603·43-s − 0.598·45-s − 1.13·47-s + 0.662·49-s + 0.345·51-s − 0.737·53-s + 0.510·55-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 | \( 1 + 4.01T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 0.942T + 11T^{2} \) |
| 13 | \( 1 + 0.0524T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 6.45T + 19T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 - 3.95T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 + 2.49T + 67T^{2} \) |
| 71 | \( 1 - 3.47T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 5.58T + 79T^{2} \) |
| 83 | \( 1 + 6.10T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 4.98T + 97T^{2} \) |
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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
−7.73434807587309712420192672337, −7.26454127084855206715813622022, −6.48764960112485770583874886598, −5.58525620490046462036626848572, −4.55001192517843629690277352340, −3.90844136799464090212766440168, −3.15636945108994463480548913567, −2.83733697042501530409433191032, −1.08996770885750032809396032239, 0,
1.08996770885750032809396032239, 2.83733697042501530409433191032, 3.15636945108994463480548913567, 3.90844136799464090212766440168, 4.55001192517843629690277352340, 5.58525620490046462036626848572, 6.48764960112485770583874886598, 7.26454127084855206715813622022, 7.73434807587309712420192672337
