| L(s) = 1 | − 2.11·2-s + 2.70·3-s + 2.46·4-s + 2.42·5-s − 5.71·6-s + 1.36·7-s − 0.977·8-s + 4.31·9-s − 5.12·10-s − 5.83·11-s + 6.65·12-s + 3.94·13-s − 2.88·14-s + 6.56·15-s − 2.86·16-s − 0.443·17-s − 9.11·18-s + 4.49·19-s + 5.97·20-s + 3.69·21-s + 12.3·22-s + 5.11·23-s − 2.64·24-s + 0.896·25-s − 8.32·26-s + 3.54·27-s + 3.36·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.56·3-s + 1.23·4-s + 1.08·5-s − 2.33·6-s + 0.516·7-s − 0.345·8-s + 1.43·9-s − 1.62·10-s − 1.76·11-s + 1.92·12-s + 1.09·13-s − 0.771·14-s + 1.69·15-s − 0.715·16-s − 0.107·17-s − 2.14·18-s + 1.03·19-s + 1.33·20-s + 0.805·21-s + 2.62·22-s + 1.06·23-s − 0.539·24-s + 0.179·25-s − 1.63·26-s + 0.683·27-s + 0.635·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.179579118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.179579118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 3 | \( 1 - 2.70T + 3T^{2} \) |
| 5 | \( 1 - 2.42T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 + 5.83T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 + 0.443T + 17T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 - 8.05T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 41 | \( 1 + 0.332T + 41T^{2} \) |
| 43 | \( 1 + 3.72T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 1.12T + 71T^{2} \) |
| 73 | \( 1 + 6.62T + 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 + 0.226T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.582907056940355013571126894692, −7.952605982292627401457624529794, −7.48508588396876698943372076437, −6.61870302189417625794941442024, −5.51783533407694472831056649690, −4.72883628744009334963114227046, −3.33543227839991485512758074809, −2.58844283851761796183320036180, −1.92352937296735810841445244773, −1.04325606947250498285760470523,
1.04325606947250498285760470523, 1.92352937296735810841445244773, 2.58844283851761796183320036180, 3.33543227839991485512758074809, 4.72883628744009334963114227046, 5.51783533407694472831056649690, 6.61870302189417625794941442024, 7.48508588396876698943372076437, 7.952605982292627401457624529794, 8.582907056940355013571126894692
