| L(s) = 1 | + 2-s + 2.30·3-s + 4-s + 2.30·6-s + 0.697·7-s + 8-s + 2.30·9-s + 0.697·11-s + 2.30·12-s + 0.697·14-s + 16-s − 2.90·17-s + 2.30·18-s + 0.394·19-s + 1.60·21-s + 0.697·22-s + 5.60·23-s + 2.30·24-s − 1.60·27-s + 0.697·28-s + 3.30·29-s + 8.60·31-s + 32-s + 1.60·33-s − 2.90·34-s + 2.30·36-s − 3.69·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.940·6-s + 0.263·7-s + 0.353·8-s + 0.767·9-s + 0.210·11-s + 0.664·12-s + 0.186·14-s + 0.250·16-s − 0.705·17-s + 0.542·18-s + 0.0904·19-s + 0.350·21-s + 0.148·22-s + 1.16·23-s + 0.470·24-s − 0.308·27-s + 0.131·28-s + 0.613·29-s + 1.54·31-s + 0.176·32-s + 0.279·33-s − 0.498·34-s + 0.383·36-s − 0.607·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.017058713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.017058713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 - 0.697T + 7T^{2} \) |
| 11 | \( 1 - 0.697T + 11T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 19 | \( 1 - 0.394T + 19T^{2} \) |
| 23 | \( 1 - 5.60T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 - 8.60T + 31T^{2} \) |
| 37 | \( 1 + 3.69T + 37T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 + 8.51T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 9.30T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 0.0916T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.954150554793730421546752127438, −6.85498103725003740869162639337, −6.75501272397092409204542361340, −5.54527113704774568523771377036, −4.88912277462681792457354010273, −4.11519216671259583496667582512, −3.47372446148312811818296624039, −2.69005565336692460095466983882, −2.16675829437366010754978288239, −1.05329000946807033464395479929,
1.05329000946807033464395479929, 2.16675829437366010754978288239, 2.69005565336692460095466983882, 3.47372446148312811818296624039, 4.11519216671259583496667582512, 4.88912277462681792457354010273, 5.54527113704774568523771377036, 6.75501272397092409204542361340, 6.85498103725003740869162639337, 7.954150554793730421546752127438
