| L(s) = 1 | − 3-s + 3·7-s − 2·9-s + 2·11-s + 13-s + 5·17-s + 19-s − 3·21-s − 23-s + 5·27-s + 3·29-s − 4·31-s − 2·33-s + 2·37-s − 39-s − 8·41-s + 8·43-s − 8·47-s + 2·49-s − 5·51-s + 9·53-s − 57-s + 59-s − 14·61-s − 6·63-s − 13·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s + 1.21·17-s + 0.229·19-s − 0.654·21-s − 0.208·23-s + 0.962·27-s + 0.557·29-s − 0.718·31-s − 0.348·33-s + 0.328·37-s − 0.160·39-s − 1.24·41-s + 1.21·43-s − 1.16·47-s + 2/7·49-s − 0.700·51-s + 1.23·53-s − 0.132·57-s + 0.130·59-s − 1.79·61-s − 0.755·63-s − 1.58·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.12563816643701, −14.92382286677239, −14.21943863010100, −14.02998844386694, −13.35386987793128, −12.51949370281869, −12.07726228555158, −11.58212582485547, −11.32457313520712, −10.56514642698179, −10.22394363483280, −9.404644236761159, −8.783632882391516, −8.355077383778564, −7.697025375406851, −7.234409518936245, −6.368886740602586, −5.868747880892866, −5.359334091179485, −4.773652789057065, −4.134562883775517, −3.325560048849139, −2.685269580800644, −1.579414869593928, −1.171327912285865, 0,
1.171327912285865, 1.579414869593928, 2.685269580800644, 3.325560048849139, 4.134562883775517, 4.773652789057065, 5.359334091179485, 5.868747880892866, 6.368886740602586, 7.234409518936245, 7.697025375406851, 8.355077383778564, 8.783632882391516, 9.404644236761159, 10.22394363483280, 10.56514642698179, 11.32457313520712, 11.58212582485547, 12.07726228555158, 12.51949370281869, 13.35386987793128, 14.02998844386694, 14.21943863010100, 14.92382286677239, 15.12563816643701
