| L(s) = 1 | − 5.90e4·3-s − 1.12e7·5-s − 2.81e8·7-s + 3.48e9·9-s + 3.61e10·11-s − 4.49e11·13-s + 6.65e11·15-s + 2.12e12·17-s + 4.60e12·19-s + 1.66e13·21-s − 9.50e13·23-s − 3.49e14·25-s − 2.05e14·27-s − 2.24e15·29-s + 3.15e15·31-s − 2.13e15·33-s + 3.17e15·35-s − 1.81e16·37-s + 2.65e16·39-s − 1.69e17·41-s + 1.58e17·43-s − 3.92e16·45-s + 1.34e17·47-s − 4.79e17·49-s − 1.25e17·51-s − 1.56e16·53-s − 4.07e17·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.516·5-s − 0.377·7-s + 1/3·9-s + 0.420·11-s − 0.903·13-s + 0.297·15-s + 0.255·17-s + 0.172·19-s + 0.217·21-s − 0.478·23-s − 0.733·25-s − 0.192·27-s − 0.991·29-s + 0.691·31-s − 0.242·33-s + 0.194·35-s − 0.621·37-s + 0.521·39-s − 1.97·41-s + 1.12·43-s − 0.172·45-s + 0.373·47-s − 0.857·49-s − 0.147·51-s − 0.0122·53-s − 0.216·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.7943477106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7943477106\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{10} T \) |
| good | 5 | \( 1 + 2253618 p T + p^{21} T^{2} \) |
| 7 | \( 1 + 40273448 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 36172082484 T + p^{21} T^{2} \) |
| 13 | \( 1 + 34546044490 p T + p^{21} T^{2} \) |
| 17 | \( 1 - 124815222738 p T + p^{21} T^{2} \) |
| 19 | \( 1 - 242600328100 p T + p^{21} T^{2} \) |
| 23 | \( 1 + 95095276921656 T + p^{21} T^{2} \) |
| 29 | \( 1 + 77439392529354 p T + p^{21} T^{2} \) |
| 31 | \( 1 - 3155693201792656 T + p^{21} T^{2} \) |
| 37 | \( 1 + 18178503074861482 T + p^{21} T^{2} \) |
| 41 | \( 1 + 169649739387485910 T + p^{21} T^{2} \) |
| 43 | \( 1 - 158968551608988244 T + p^{21} T^{2} \) |
| 47 | \( 1 - 134697468442682736 T + p^{21} T^{2} \) |
| 53 | \( 1 + 15637375269722538 T + p^{21} T^{2} \) |
| 59 | \( 1 + 2977241337691499484 T + p^{21} T^{2} \) |
| 61 | \( 1 - 3603855625679330702 T + p^{21} T^{2} \) |
| 67 | \( 1 + 21066199531967164004 T + p^{21} T^{2} \) |
| 71 | \( 1 + 21980089544074358760 T + p^{21} T^{2} \) |
| 73 | \( 1 + 17054415965500339222 T + p^{21} T^{2} \) |
| 79 | \( 1 - \)\(11\!\cdots\!52\)\( T + p^{21} T^{2} \) |
| 83 | \( 1 - 96628520442403345644 T + p^{21} T^{2} \) |
| 89 | \( 1 - 60427571095732966650 T + p^{21} T^{2} \) |
| 97 | \( 1 + \)\(40\!\cdots\!98\)\( T + p^{21} T^{2} \) |
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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
−11.67409132485051322154279197855, −10.36652225250716813593586926945, −9.366807472405104822155575883824, −7.88486298915501931754915962936, −6.86312682489487977002013138974, −5.67520711492847929544933250341, −4.46120320683690130410420877426, −3.32808476389794604549016631351, −1.81584871455292731174272150228, −0.40967457952288880523679252436,
0.40967457952288880523679252436, 1.81584871455292731174272150228, 3.32808476389794604549016631351, 4.46120320683690130410420877426, 5.67520711492847929544933250341, 6.86312682489487977002013138974, 7.88486298915501931754915962936, 9.366807472405104822155575883824, 10.36652225250716813593586926945, 11.67409132485051322154279197855
