| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 9-s + 10·11-s + 5·16-s + 2·18-s + 20·22-s − 8·23-s + 12·29-s + 6·32-s + 3·36-s − 8·37-s − 16·43-s + 30·44-s − 16·46-s + 8·53-s + 24·58-s + 7·64-s + 10·67-s + 12·71-s + 4·72-s − 16·74-s + 20·79-s − 8·81-s − 32·86-s + 40·88-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 1/3·9-s + 3.01·11-s + 5/4·16-s + 0.471·18-s + 4.26·22-s − 1.66·23-s + 2.22·29-s + 1.06·32-s + 1/2·36-s − 1.31·37-s − 2.43·43-s + 4.52·44-s − 2.35·46-s + 1.09·53-s + 3.15·58-s + 7/8·64-s + 1.22·67-s + 1.42·71-s + 0.471·72-s − 1.85·74-s + 2.25·79-s − 8/9·81-s − 3.45·86-s + 4.26·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.553764546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.553764546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 139 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 159 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.958249969269538504758571391809, −8.804317404253944380183838237818, −8.238453358643676074796464535368, −8.123206020785911224047205617876, −7.30666972105460747884646218972, −6.94564511451168142471036958807, −6.74331653574359212690083389660, −6.30494996008142518305942590146, −6.20378140078212591614595264169, −5.68296685974423073062926668570, −4.90123892865172734409730900354, −4.84853051652488384264655867416, −4.32142764870998178576054646880, −3.78676284117144683643186928183, −3.58508012431733577971752563188, −3.35630122746138909239718552725, −2.38289110161975670974100267321, −1.98028389622359504742818949135, −1.44168099916859677764667416238, −0.875391699215869102345774678715,
0.875391699215869102345774678715, 1.44168099916859677764667416238, 1.98028389622359504742818949135, 2.38289110161975670974100267321, 3.35630122746138909239718552725, 3.58508012431733577971752563188, 3.78676284117144683643186928183, 4.32142764870998178576054646880, 4.84853051652488384264655867416, 4.90123892865172734409730900354, 5.68296685974423073062926668570, 6.20378140078212591614595264169, 6.30494996008142518305942590146, 6.74331653574359212690083389660, 6.94564511451168142471036958807, 7.30666972105460747884646218972, 8.123206020785911224047205617876, 8.238453358643676074796464535368, 8.804317404253944380183838237818, 8.958249969269538504758571391809
