| L(s) = 1 | − 10·5-s + 7·9-s + 24·13-s + 28·17-s + 25·25-s + 48·29-s − 102·37-s + 80·41-s − 70·45-s + 54·49-s − 92·53-s + 88·61-s − 240·65-s + 88·73-s − 32·81-s − 280·85-s − 178·89-s + 370·97-s − 132·101-s + 364·109-s + 314·113-s + 168·117-s − 11·121-s + 250·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·5-s + 7/9·9-s + 1.84·13-s + 1.64·17-s + 25-s + 1.65·29-s − 2.75·37-s + 1.95·41-s − 1.55·45-s + 1.10·49-s − 1.73·53-s + 1.44·61-s − 3.69·65-s + 1.20·73-s − 0.395·81-s − 3.29·85-s − 2·89-s + 3.81·97-s − 1.30·101-s + 3.33·109-s + 2.77·113-s + 1.43·117-s − 0.0909·121-s + 2·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.098978576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.098978576\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )( 1 + 5 T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 54 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1047 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 591 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 51 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 134 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4206 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5631 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 4127 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6903 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12086 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2106 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 185 T + p^{2} T^{2} )^{2} \) |
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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
−10.54094764262135898028814079720, −9.989946706439953340126475525675, −9.807089051928401843406539613529, −8.892381058138600834257333797296, −8.696032581055963454742862323237, −8.174679397039434925181927841747, −7.948321026391900012241550503650, −7.44324416771989257947224299819, −7.09183017479897049369334981027, −6.63033594887199001776033553500, −5.86578561939854134785483228241, −5.69660184191224083411928012484, −4.72215167866739939236819105737, −4.47376745861606238728966597585, −3.78886951960443329061591190523, −3.46599465655791031829510241732, −3.24193971221921742996838967101, −2.02514680299661315802694551036, −1.13120180409472766606793131913, −0.63304703990025272971206086134,
0.63304703990025272971206086134, 1.13120180409472766606793131913, 2.02514680299661315802694551036, 3.24193971221921742996838967101, 3.46599465655791031829510241732, 3.78886951960443329061591190523, 4.47376745861606238728966597585, 4.72215167866739939236819105737, 5.69660184191224083411928012484, 5.86578561939854134785483228241, 6.63033594887199001776033553500, 7.09183017479897049369334981027, 7.44324416771989257947224299819, 7.948321026391900012241550503650, 8.174679397039434925181927841747, 8.696032581055963454742862323237, 8.892381058138600834257333797296, 9.807089051928401843406539613529, 9.989946706439953340126475525675, 10.54094764262135898028814079720
