| L(s) = 1 | − 4-s − 10·7-s − 6·9-s + 6·11-s + 16-s + 10·28-s + 6·36-s − 2·37-s + 10·41-s − 6·44-s + 16·47-s + 61·49-s + 18·53-s + 60·63-s − 64-s − 4·67-s + 4·71-s − 12·73-s − 60·77-s + 27·81-s + 24·83-s − 36·99-s + 16·101-s + 28·107-s − 10·112-s + 5·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3.77·7-s − 2·9-s + 1.80·11-s + 1/4·16-s + 1.88·28-s + 36-s − 0.328·37-s + 1.56·41-s − 0.904·44-s + 2.33·47-s + 61/7·49-s + 2.47·53-s + 7.55·63-s − 1/8·64-s − 0.488·67-s + 0.474·71-s − 1.40·73-s − 6.83·77-s + 3·81-s + 2.63·83-s − 3.61·99-s + 1.59·101-s + 2.70·107-s − 0.944·112-s + 5/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8639565017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8639565017\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
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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
−9.197123183672336145875615392950, −9.097026969224528464996828387806, −8.808193409323573546882328123944, −8.783137616420703307066309163680, −7.79270653217172769924317590940, −7.35003113111458534954330187014, −6.97175650388843926667013306918, −6.48226409324720804246802830265, −6.24801332737327098645897784963, −5.90017560759772518586108884127, −5.75767910357374920338086374482, −5.14118629331863442186104103860, −4.18729521393121976565542039002, −3.88815196190527950712262542963, −3.52573128384839910438874423532, −3.24171716526628393697770844992, −2.49763506358573067281624642719, −2.49182816879483593459715922638, −0.804147687839415326368442514654, −0.49684841120786985915588843231,
0.49684841120786985915588843231, 0.804147687839415326368442514654, 2.49182816879483593459715922638, 2.49763506358573067281624642719, 3.24171716526628393697770844992, 3.52573128384839910438874423532, 3.88815196190527950712262542963, 4.18729521393121976565542039002, 5.14118629331863442186104103860, 5.75767910357374920338086374482, 5.90017560759772518586108884127, 6.24801332737327098645897784963, 6.48226409324720804246802830265, 6.97175650388843926667013306918, 7.35003113111458534954330187014, 7.79270653217172769924317590940, 8.783137616420703307066309163680, 8.808193409323573546882328123944, 9.097026969224528464996828387806, 9.197123183672336145875615392950
