| L(s) = 1 | + 8.16·5-s − 37.8·11-s − 39.9·13-s + 9.93·17-s + 90.4·19-s − 118.·23-s − 58.2·25-s + 78.4·29-s − 92.0·31-s + 332.·37-s + 71.7·41-s − 115.·43-s + 307.·47-s + 403.·53-s − 308.·55-s + 593.·59-s − 333.·61-s − 326.·65-s − 743.·67-s + 728.·71-s + 801.·73-s + 1.06e3·79-s + 906.·83-s + 81.1·85-s + 1.11e3·89-s + 738.·95-s − 1.48e3·97-s + ⋯ |
| L(s) = 1 | + 0.730·5-s − 1.03·11-s − 0.851·13-s + 0.141·17-s + 1.09·19-s − 1.07·23-s − 0.466·25-s + 0.502·29-s − 0.533·31-s + 1.47·37-s + 0.273·41-s − 0.411·43-s + 0.955·47-s + 1.04·53-s − 0.757·55-s + 1.31·59-s − 0.699·61-s − 0.622·65-s − 1.35·67-s + 1.21·71-s + 1.28·73-s + 1.52·79-s + 1.19·83-s + 0.103·85-s + 1.32·89-s + 0.797·95-s − 1.55·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.066527043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066527043\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8.16T + 125T^{2} \) |
| 11 | \( 1 + 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.93T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 78.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 92.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 332.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 71.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 403.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 593.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 333.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 743.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 801.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 906.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.101306674294444017839167369567, −7.930384545190157595775926725737, −7.56117118589584482815949858674, −6.45781376072011445631024732460, −5.59720813655768288505918503432, −5.06292074714001884499185915277, −3.93085655362888110026282159963, −2.73387827838392153395956150467, −2.05110625121299671118286703014, −0.66116521202031232703735183071,
0.66116521202031232703735183071, 2.05110625121299671118286703014, 2.73387827838392153395956150467, 3.93085655362888110026282159963, 5.06292074714001884499185915277, 5.59720813655768288505918503432, 6.45781376072011445631024732460, 7.56117118589584482815949858674, 7.930384545190157595775926725737, 9.101306674294444017839167369567
