| L(s) = 1 | − 2.33·3-s + 4.35·5-s − 7-s + 2.45·9-s − 0.727·11-s + 6.02·13-s − 10.1·15-s − 0.705·17-s + 19-s + 2.33·21-s − 2.70·23-s + 13.9·25-s + 1.27·27-s − 6.12·29-s + 9.48·31-s + 1.69·33-s − 4.35·35-s + 2.17·37-s − 14.0·39-s + 4.90·41-s + 2.88·43-s + 10.6·45-s − 3.05·47-s + 49-s + 1.64·51-s − 7.16·53-s − 3.16·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 1.94·5-s − 0.377·7-s + 0.818·9-s − 0.219·11-s + 1.67·13-s − 2.62·15-s − 0.171·17-s + 0.229·19-s + 0.509·21-s − 0.564·23-s + 2.79·25-s + 0.245·27-s − 1.13·29-s + 1.70·31-s + 0.295·33-s − 0.736·35-s + 0.357·37-s − 2.25·39-s + 0.766·41-s + 0.439·43-s + 1.59·45-s − 0.445·47-s + 0.142·49-s + 0.230·51-s − 0.983·53-s − 0.427·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.007057286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007057286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.33T + 3T^{2} \) |
| 5 | \( 1 - 4.35T + 5T^{2} \) |
| 11 | \( 1 + 0.727T + 11T^{2} \) |
| 13 | \( 1 - 6.02T + 13T^{2} \) |
| 17 | \( 1 + 0.705T + 17T^{2} \) |
| 23 | \( 1 + 2.70T + 23T^{2} \) |
| 29 | \( 1 + 6.12T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 - 4.90T + 41T^{2} \) |
| 43 | \( 1 - 2.88T + 43T^{2} \) |
| 47 | \( 1 + 3.05T + 47T^{2} \) |
| 53 | \( 1 + 7.16T + 53T^{2} \) |
| 59 | \( 1 + 5.08T + 59T^{2} \) |
| 61 | \( 1 + 8.14T + 61T^{2} \) |
| 67 | \( 1 + 7.65T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 7.93T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 9.13T + 89T^{2} \) |
| 97 | \( 1 + 5.08T + 97T^{2} \) |
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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
−7.67539305473076930454004017245, −6.54403979447090592905211195616, −6.21643611345803259387250118380, −5.96708932105999823921656176831, −5.22767945776238232300041250315, −4.58478921053924323919053877985, −3.45198030044210460894640093518, −2.49115101018975378095105274438, −1.55800097406577375954724829360, −0.795364234252009560588220836194,
0.795364234252009560588220836194, 1.55800097406577375954724829360, 2.49115101018975378095105274438, 3.45198030044210460894640093518, 4.58478921053924323919053877985, 5.22767945776238232300041250315, 5.96708932105999823921656176831, 6.21643611345803259387250118380, 6.54403979447090592905211195616, 7.67539305473076930454004017245
