L(s) = 1 | − 2.69·3-s + 4.28·9-s + 0.919·11-s + 5.24·13-s + 4.25·17-s + 1.83·19-s − 8.78·23-s − 3.46·27-s + 6.71·29-s − 4.64·31-s − 2.48·33-s − 1.42·37-s − 14.1·39-s − 7.35·41-s − 9.80·43-s − 6.80·47-s − 11.4·51-s − 11.2·53-s − 4.96·57-s + 11.3·59-s − 9.64·61-s − 2.78·67-s + 23.6·69-s + 11.8·71-s + 5.11·73-s + 0.727·79-s − 3.50·81-s + ⋯ |
L(s) = 1 | − 1.55·3-s + 1.42·9-s + 0.277·11-s + 1.45·13-s + 1.03·17-s + 0.422·19-s − 1.83·23-s − 0.666·27-s + 1.24·29-s − 0.833·31-s − 0.432·33-s − 0.233·37-s − 2.26·39-s − 1.14·41-s − 1.49·43-s − 0.992·47-s − 1.60·51-s − 1.54·53-s − 0.657·57-s + 1.47·59-s − 1.23·61-s − 0.340·67-s + 2.85·69-s + 1.40·71-s + 0.598·73-s + 0.0818·79-s − 0.389·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 11 | \( 1 - 0.919T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 + 8.78T + 23T^{2} \) |
| 29 | \( 1 - 6.71T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 + 7.35T + 41T^{2} \) |
| 43 | \( 1 + 9.80T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.64T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 - 0.727T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 + 0.963T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.10315262642980619530344187753, −6.31875323483475148755536205834, −6.14936156243396906212926488692, −5.31534944183581175638275025509, −4.81150061698239449107378155803, −3.82132625246977680582624952390, −3.30552136656473881136606739185, −1.77614610865345883368166791147, −1.11024909154849437743405045192, 0,
1.11024909154849437743405045192, 1.77614610865345883368166791147, 3.30552136656473881136606739185, 3.82132625246977680582624952390, 4.81150061698239449107378155803, 5.31534944183581175638275025509, 6.14936156243396906212926488692, 6.31875323483475148755536205834, 7.10315262642980619530344187753