| L(s) = 1 | + 4·2-s + 2·3-s + 12·4-s + 8·6-s + 32·8-s − 5·9-s − 68·11-s + 24·12-s + 52·13-s + 80·16-s − 164·17-s − 20·18-s − 232·19-s − 272·22-s + 198·23-s + 64·24-s + 208·26-s + 26·27-s − 18·29-s + 196·31-s + 192·32-s − 136·33-s − 656·34-s − 60·36-s − 160·37-s − 928·38-s + 104·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.384·3-s + 3/2·4-s + 0.544·6-s + 1.41·8-s − 0.185·9-s − 1.86·11-s + 0.577·12-s + 1.10·13-s + 5/4·16-s − 2.33·17-s − 0.261·18-s − 2.80·19-s − 2.63·22-s + 1.79·23-s + 0.544·24-s + 1.56·26-s + 0.185·27-s − 0.115·29-s + 1.13·31-s + 1.06·32-s − 0.717·33-s − 3.30·34-s − 0.277·36-s − 0.710·37-s − 3.96·38-s + 0.427·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + p^{2} T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 68 T + 3634 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 p T + 4334 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 164 T + 13606 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 232 T + 26438 T^{2} + 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 198 T + 23785 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 18 T + 33955 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 196 T + 27786 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 160 T + 101082 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 62 T + 72379 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 198 T + 140065 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 164 T + 211426 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 40 T + 5570 p T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 80 T - 85178 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 174 T + 105307 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1054 T + 672761 T^{2} - 1054 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 832 T + 815278 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 820 T + 840150 T^{2} + 820 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 576 T + 880606 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 298 T + 423841 T^{2} - 298 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 182 T + 1152523 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 892 T + 1918278 T^{2} + 892 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.354658952835388046202167310952, −8.187673571189655244603078824152, −7.57465080375996799891289318412, −6.90714434106270346087798044726, −6.71740962413932961287163702057, −6.63680348611665449225938763706, −5.88988120161379085865889557877, −5.75158672978376704463442745305, −5.13550381135057044759596539357, −4.66310350134555608905833301702, −4.40455092795234646665932757312, −4.24582737496812417039271302157, −3.41874288233491153768869742762, −3.10075617031719686066641156913, −2.50468633327813329440123127166, −2.38591784962061031427253447079, −1.84380661414679459404132589639, −1.15400041232124511000347772135, 0, 0,
1.15400041232124511000347772135, 1.84380661414679459404132589639, 2.38591784962061031427253447079, 2.50468633327813329440123127166, 3.10075617031719686066641156913, 3.41874288233491153768869742762, 4.24582737496812417039271302157, 4.40455092795234646665932757312, 4.66310350134555608905833301702, 5.13550381135057044759596539357, 5.75158672978376704463442745305, 5.88988120161379085865889557877, 6.63680348611665449225938763706, 6.71740962413932961287163702057, 6.90714434106270346087798044726, 7.57465080375996799891289318412, 8.187673571189655244603078824152, 8.354658952835388046202167310952
