| L(s) = 1 | − 2-s − 4-s + 3.56·5-s + 3·8-s − 3.56·10-s + 3.56·11-s − 3.56·13-s − 16-s − 5.12·19-s − 3.56·20-s − 3.56·22-s − 23-s + 7.68·25-s + 3.56·26-s + 29-s − 2.43·31-s − 5·32-s + 2.43·37-s + 5.12·38-s + 10.6·40-s − 7.56·41-s − 4.24·43-s − 3.56·44-s + 46-s − 7.12·47-s − 7·49-s − 7.68·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.5·4-s + 1.59·5-s + 1.06·8-s − 1.12·10-s + 1.07·11-s − 0.987·13-s − 0.250·16-s − 1.17·19-s − 0.796·20-s − 0.759·22-s − 0.208·23-s + 1.53·25-s + 0.698·26-s + 0.185·29-s − 0.437·31-s − 0.883·32-s + 0.400·37-s + 0.831·38-s + 1.68·40-s − 1.18·41-s − 0.647·43-s − 0.536·44-s + 0.147·46-s − 1.03·47-s − 49-s − 1.08·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 9.56T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 7.12T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−7.910580099573488211738702404049, −6.92013375058656279810116379145, −6.43397134591433395658737814711, −5.62316983099132386787689076378, −4.84517885518254766041930415841, −4.23932622004694222893386678951, −3.05030636398238243356873778693, −1.91800926960019710086621260958, −1.47155029626420119034972834326, 0,
1.47155029626420119034972834326, 1.91800926960019710086621260958, 3.05030636398238243356873778693, 4.23932622004694222893386678951, 4.84517885518254766041930415841, 5.62316983099132386787689076378, 6.43397134591433395658737814711, 6.92013375058656279810116379145, 7.910580099573488211738702404049
