| L(s) = 1 | − 1.02e3·2-s + 7.16e4·3-s + 1.04e6·4-s − 2.86e7·5-s − 7.33e7·6-s − 8.53e8·7-s − 1.07e9·8-s − 5.33e9·9-s + 2.93e10·10-s + 8.67e10·11-s + 7.50e10·12-s − 8.95e11·13-s + 8.73e11·14-s − 2.05e12·15-s + 1.09e12·16-s + 3.25e12·17-s + 5.46e12·18-s + 2.30e13·19-s − 3.00e13·20-s − 6.10e13·21-s − 8.88e13·22-s + 1.46e14·23-s − 7.68e13·24-s + 3.46e14·25-s + 9.16e14·26-s − 1.13e15·27-s − 8.94e14·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.700·3-s + 1/2·4-s − 1.31·5-s − 0.495·6-s − 1.14·7-s − 0.353·8-s − 0.509·9-s + 0.929·10-s + 1.00·11-s + 0.350·12-s − 1.80·13-s + 0.807·14-s − 0.919·15-s + 1/4·16-s + 0.391·17-s + 0.360·18-s + 0.861·19-s − 0.657·20-s − 0.799·21-s − 0.712·22-s + 0.737·23-s − 0.247·24-s + 0.726·25-s + 1.27·26-s − 1.05·27-s − 0.570·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{10} T \) |
| good | 3 | \( 1 - 884 p^{4} T + p^{21} T^{2} \) |
| 5 | \( 1 + 5738754 p T + p^{21} T^{2} \) |
| 7 | \( 1 + 121886056 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 7884652692 p T + p^{21} T^{2} \) |
| 13 | \( 1 + 895323442786 T + p^{21} T^{2} \) |
| 17 | \( 1 - 191621576754 p T + p^{21} T^{2} \) |
| 19 | \( 1 - 1212235139180 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 146495714575224 T + p^{21} T^{2} \) |
| 29 | \( 1 + 734051633521170 T + p^{21} T^{2} \) |
| 31 | \( 1 + 3146664162057568 T + p^{21} T^{2} \) |
| 37 | \( 1 + 12963813600992362 T + p^{21} T^{2} \) |
| 41 | \( 1 - 45714648841476042 T + p^{21} T^{2} \) |
| 43 | \( 1 + 24073607797047556 T + p^{21} T^{2} \) |
| 47 | \( 1 + 449991905173684752 T + p^{21} T^{2} \) |
| 53 | \( 1 - 2064837217091540454 T + p^{21} T^{2} \) |
| 59 | \( 1 + 3780497099978396340 T + p^{21} T^{2} \) |
| 61 | \( 1 + 7619813346829729138 T + p^{21} T^{2} \) |
| 67 | \( 1 + 18791158016925310732 T + p^{21} T^{2} \) |
| 71 | \( 1 + 4526486567453771928 T + p^{21} T^{2} \) |
| 73 | \( 1 + 25571455286910443926 T + p^{21} T^{2} \) |
| 79 | \( 1 - 99336442530925070480 T + p^{21} T^{2} \) |
| 83 | \( 1 - 2958180217887529284 T + p^{21} T^{2} \) |
| 89 | \( 1 - \)\(11\!\cdots\!90\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 + \)\(56\!\cdots\!02\)\( T + p^{21} T^{2} \) |
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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
−22.64029125098420703132144847743, −19.83038971548078417888817778811, −19.38773535117066641148860531955, −16.65789742081195286901250748112, −14.88707859578254233336740922202, −11.97275245883018303327935557123, −9.330335973146124666993638438675, −7.40556770786594456447332544948, −3.21390523418769017977825296607, 0,
3.21390523418769017977825296607, 7.40556770786594456447332544948, 9.330335973146124666993638438675, 11.97275245883018303327935557123, 14.88707859578254233336740922202, 16.65789742081195286901250748112, 19.38773535117066641148860531955, 19.83038971548078417888817778811, 22.64029125098420703132144847743
