L(s) = 1 | + 2.82·5-s − 6·11-s + 5.65·13-s + 1.41·17-s + 4.24·19-s − 4·23-s + 3.00·25-s + 6·29-s − 2.82·31-s + 2·37-s − 1.41·41-s + 10·43-s − 2.82·47-s + 2·53-s − 16.9·55-s + 1.41·59-s + 8.48·61-s + 16.0·65-s + 4·67-s + 12·71-s + 9.89·73-s − 4·79-s + 1.41·83-s + 4.00·85-s − 4.24·89-s + 12·95-s − 12.7·97-s + ⋯ |
L(s) = 1 | + 1.26·5-s − 1.80·11-s + 1.56·13-s + 0.342·17-s + 0.973·19-s − 0.834·23-s + 0.600·25-s + 1.11·29-s − 0.508·31-s + 0.328·37-s − 0.220·41-s + 1.52·43-s − 0.412·47-s + 0.274·53-s − 2.28·55-s + 0.184·59-s + 1.08·61-s + 1.98·65-s + 0.488·67-s + 1.42·71-s + 1.15·73-s − 0.450·79-s + 0.155·83-s + 0.433·85-s − 0.449·89-s + 1.23·95-s − 1.29·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.436658852\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.436658852\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 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
−8.417785025089098975517243289351, −8.022973242666175078235766606173, −7.05048551759444525519261316964, −6.10075081561229970736180324067, −5.63252449148214209221580864945, −5.04024085589718567933654509490, −3.82948899268086455961308645554, −2.85799735933844640095495338182, −2.08263817056232641145314108233, −0.948427762321686055952858963498,
0.948427762321686055952858963498, 2.08263817056232641145314108233, 2.85799735933844640095495338182, 3.82948899268086455961308645554, 5.04024085589718567933654509490, 5.63252449148214209221580864945, 6.10075081561229970736180324067, 7.05048551759444525519261316964, 8.022973242666175078235766606173, 8.417785025089098975517243289351