L(s) = 1 | + 5·5-s + 6.21·7-s − 28.3·11-s − 8.60·13-s − 90.1·17-s + 114.·19-s − 48.4·23-s + 25·25-s + 305.·29-s + 93.5·31-s + 31.0·35-s − 282.·37-s − 28.7·41-s − 354.·43-s − 522.·47-s − 304.·49-s − 66.9·53-s − 141.·55-s − 7.63·59-s + 9.41·61-s − 43.0·65-s − 494.·67-s − 560.·71-s + 1.11e3·73-s − 176.·77-s + 1.04e3·79-s + 45.5·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.335·7-s − 0.777·11-s − 0.183·13-s − 1.28·17-s + 1.37·19-s − 0.438·23-s + 0.200·25-s + 1.95·29-s + 0.542·31-s + 0.150·35-s − 1.25·37-s − 0.109·41-s − 1.25·43-s − 1.62·47-s − 0.887·49-s − 0.173·53-s − 0.347·55-s − 0.0168·59-s + 0.0197·61-s − 0.0821·65-s − 0.901·67-s − 0.937·71-s + 1.79·73-s − 0.260·77-s + 1.48·79-s + 0.0601·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
good | 7 | \( 1 - 6.21T + 343T^{2} \) |
| 11 | \( 1 + 28.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.60T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 48.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 305.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 93.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 282.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 28.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 522.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 66.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 7.63T + 2.05e5T^{2} \) |
| 61 | \( 1 - 9.41T + 2.26e5T^{2} \) |
| 67 | \( 1 + 494.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 560.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 45.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 357.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 120.T + 9.12e5T^{2} \) |
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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
−8.512688786562910822058899111181, −7.992988409909124229167592250982, −6.92698301228756253723847483261, −6.30049976587366383391185463007, −5.11325680993502854980924483048, −4.75744482642598091176716150796, −3.35056355666252147751378211434, −2.45078976799040588997814903216, −1.39229892139978482185168817521, 0,
1.39229892139978482185168817521, 2.45078976799040588997814903216, 3.35056355666252147751378211434, 4.75744482642598091176716150796, 5.11325680993502854980924483048, 6.30049976587366383391185463007, 6.92698301228756253723847483261, 7.992988409909124229167592250982, 8.512688786562910822058899111181