| L(s) = 1 | − 2.90·3-s + 5-s + 2.18·7-s + 5.43·9-s − 6.16·13-s − 2.90·15-s − 7.12·17-s − 6.35·19-s − 6.34·21-s + 2.17·23-s + 25-s − 7.08·27-s + 3.19·29-s + 4.39·31-s + 2.18·35-s + 5.29·37-s + 17.8·39-s − 3.28·41-s + 6.83·43-s + 5.43·45-s − 4.30·47-s − 2.23·49-s + 20.7·51-s − 10.6·53-s + 18.4·57-s + 13.7·59-s − 7.35·61-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.447·5-s + 0.825·7-s + 1.81·9-s − 1.70·13-s − 0.750·15-s − 1.72·17-s − 1.45·19-s − 1.38·21-s + 0.454·23-s + 0.200·25-s − 1.36·27-s + 0.592·29-s + 0.789·31-s + 0.369·35-s + 0.870·37-s + 2.86·39-s − 0.513·41-s + 1.04·43-s + 0.810·45-s − 0.627·47-s − 0.319·49-s + 2.89·51-s − 1.45·53-s + 2.44·57-s + 1.79·59-s − 0.941·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7630940270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7630940270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 13 | \( 1 + 6.16T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 - 4.39T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 + 3.28T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 + 4.30T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 7.35T + 61T^{2} \) |
| 67 | \( 1 + 7.67T + 67T^{2} \) |
| 71 | \( 1 - 5.74T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 - 8.11T + 79T^{2} \) |
| 83 | \( 1 + 4.22T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 3.27T + 97T^{2} \) |
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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
−8.190203278888162715474667599286, −7.33880430227285786555781492088, −6.51692006496618885944381963493, −6.29421050865654906853954336968, −5.16885457024742393920711577678, −4.73187030169045684492707735112, −4.33451746826290823119039642656, −2.55900537997964834910499636542, −1.82875798800132782955081270674, −0.51205311569964991798862649159,
0.51205311569964991798862649159, 1.82875798800132782955081270674, 2.55900537997964834910499636542, 4.33451746826290823119039642656, 4.73187030169045684492707735112, 5.16885457024742393920711577678, 6.29421050865654906853954336968, 6.51692006496618885944381963493, 7.33880430227285786555781492088, 8.190203278888162715474667599286
