L(s) = 1 | − 1.41·3-s − 1.41·5-s − 0.999·9-s + 11-s + 2.00·15-s + 5.65·19-s + 2·23-s − 2.99·25-s + 5.65·27-s − 6·29-s − 4.24·31-s − 1.41·33-s − 10·37-s + 5.65·41-s + 8·43-s + 1.41·45-s − 9.89·47-s − 1.41·55-s − 8.00·57-s + 9.89·59-s + 8.48·61-s − 2·67-s − 2.82·69-s − 6·71-s − 8.48·73-s + 4.24·75-s + 8·79-s + ⋯ |
L(s) = 1 | − 0.816·3-s − 0.632·5-s − 0.333·9-s + 0.301·11-s + 0.516·15-s + 1.29·19-s + 0.417·23-s − 0.599·25-s + 1.08·27-s − 1.11·29-s − 0.762·31-s − 0.246·33-s − 1.64·37-s + 0.883·41-s + 1.21·43-s + 0.210·45-s − 1.44·47-s − 0.190·55-s − 1.05·57-s + 1.28·59-s + 1.08·61-s − 0.244·67-s − 0.340·69-s − 0.712·71-s − 0.993·73-s + 0.489·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.39514258235472996323322965775, −6.78550584984039047651111119572, −5.91584300280621359594080646618, −5.41381966379873278722077099955, −4.78795303233504913360003726568, −3.77802792353560886074095693893, −3.30380259677598838829692816105, −2.14679084491488806263380700082, −1.00096275034805857195732354190, 0,
1.00096275034805857195732354190, 2.14679084491488806263380700082, 3.30380259677598838829692816105, 3.77802792353560886074095693893, 4.78795303233504913360003726568, 5.41381966379873278722077099955, 5.91584300280621359594080646618, 6.78550584984039047651111119572, 7.39514258235472996323322965775