L(s) = 1 | − 2-s + 2.89·3-s + 4-s − 3.80·5-s − 2.89·6-s − 7-s − 8-s + 5.36·9-s + 3.80·10-s + 0.838·11-s + 2.89·12-s − 6.40·13-s + 14-s − 11.0·15-s + 16-s + 4.08·17-s − 5.36·18-s + 4.40·19-s − 3.80·20-s − 2.89·21-s − 0.838·22-s − 0.575·23-s − 2.89·24-s + 9.51·25-s + 6.40·26-s + 6.83·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.66·3-s + 0.5·4-s − 1.70·5-s − 1.18·6-s − 0.377·7-s − 0.353·8-s + 1.78·9-s + 1.20·10-s + 0.252·11-s + 0.834·12-s − 1.77·13-s + 0.267·14-s − 2.84·15-s + 0.250·16-s + 0.989·17-s − 1.26·18-s + 1.01·19-s − 0.851·20-s − 0.631·21-s − 0.178·22-s − 0.120·23-s − 0.590·24-s + 1.90·25-s + 1.25·26-s + 1.31·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 11 | \( 1 - 0.838T + 11T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 - 4.08T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 + 0.575T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 5.08T + 31T^{2} \) |
| 37 | \( 1 - 0.240T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 - 9.84T + 43T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 + 9.62T + 59T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 + 0.717T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 0.394T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.75067360144804385706083739857, −7.35054076154990778268814445011, −7.01136176226793542315983892508, −5.53942177723757409163383975915, −4.46395445393441512250318620843, −3.79638990718397226173507320304, −3.05946540324624011207052067260, −2.60760295156596901954103135926, −1.30712565443454358254619462068, 0,
1.30712565443454358254619462068, 2.60760295156596901954103135926, 3.05946540324624011207052067260, 3.79638990718397226173507320304, 4.46395445393441512250318620843, 5.53942177723757409163383975915, 7.01136176226793542315983892508, 7.35054076154990778268814445011, 7.75067360144804385706083739857