L(s) = 1 | + 1.41·3-s + 2.82·7-s − 0.999·9-s − 0.828·11-s − 3.41·13-s − 4.82·17-s − 19-s + 4.00·21-s − 4·23-s − 5.65·27-s − 0.828·29-s − 1.17·33-s − 10.2·37-s − 4.82·39-s − 0.828·41-s − 2.82·43-s + 8.48·47-s + 1.00·49-s − 6.82·51-s − 13.0·53-s − 1.41·57-s + 2.82·59-s − 1.65·61-s − 2.82·63-s + 9.41·67-s − 5.65·69-s + 15.3·71-s + ⋯ |
L(s) = 1 | + 0.816·3-s + 1.06·7-s − 0.333·9-s − 0.249·11-s − 0.946·13-s − 1.17·17-s − 0.229·19-s + 0.872·21-s − 0.834·23-s − 1.08·27-s − 0.153·29-s − 0.203·33-s − 1.68·37-s − 0.773·39-s − 0.129·41-s − 0.431·43-s + 1.23·47-s + 0.142·49-s − 0.956·51-s − 1.79·53-s − 0.187·57-s + 0.368·59-s − 0.212·61-s − 0.356·63-s + 1.15·67-s − 0.681·69-s + 1.81·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 9.41T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 - 2.24T + 97T^{2} \) |
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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
−8.269738606703878644712072772452, −7.55194113962969876212176507192, −6.84961150136849014053366910403, −5.80609691758732056345161171967, −4.99932687992669003569647448932, −4.33551937764430817000688339613, −3.35482701674970837480485780563, −2.34836520621442876982072622037, −1.82472000436605057785289224595, 0,
1.82472000436605057785289224595, 2.34836520621442876982072622037, 3.35482701674970837480485780563, 4.33551937764430817000688339613, 4.99932687992669003569647448932, 5.80609691758732056345161171967, 6.84961150136849014053366910403, 7.55194113962969876212176507192, 8.269738606703878644712072772452