| L(s) = 1 | − 256·4-s + 1.50e4·9-s − 1.12e5·11-s + 6.55e4·16-s − 6.30e5·19-s + 7.68e6·29-s − 2.61e6·31-s − 3.84e6·36-s + 3.02e6·41-s + 2.87e7·44-s + 7.98e7·49-s − 2.24e8·59-s − 6.63e7·61-s − 1.67e7·64-s − 7.74e8·71-s + 1.61e8·76-s − 9.84e8·79-s − 1.61e8·81-s + 6.36e7·89-s − 1.68e9·99-s + 2.11e9·101-s + 2.68e9·109-s − 1.96e9·116-s + 4.74e9·121-s + 6.70e8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.763·9-s − 2.31·11-s + 1/4·16-s − 1.11·19-s + 2.01·29-s − 0.509·31-s − 0.381·36-s + 0.167·41-s + 1.15·44-s + 1.97·49-s − 2.41·59-s − 0.613·61-s − 1/8·64-s − 3.61·71-s + 0.555·76-s − 2.84·79-s − 0.416·81-s + 0.107·89-s − 1.76·99-s + 2.02·101-s + 1.82·109-s − 1.00·116-s + 2.01·121-s + 0.254·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7365245701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7365245701\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( 1 + p^{8} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 1670 p^{2} T^{2} + p^{18} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1628590 p^{2} T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 56148 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 10508474090 T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 175839286750 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 315380 T + p^{9} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3560483436910 T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3840450 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 1309408 T + p^{9} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 241372558692070 T^{2} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1512042 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 128524349851130 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2126301861536350 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6323428615910470 T^{2} + p^{18} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 112235100 T + p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 33197218 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 39681845237038390 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 387172728 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 52595678041010350 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 492101840 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 164647238232785110 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 31809510 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1066816678767158590 T^{2} + p^{18} T^{4} \) |
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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
−13.68781485271226063105839512204, −13.35279222730060506210538647871, −12.59098893486709497182235246679, −12.57518152121079988791880939437, −11.56966940314310702646563630120, −10.65459078969536679186789192866, −10.30370969622166582664606012401, −10.06066787452777894299391762156, −8.828994686734232356702640091435, −8.613331823451940328379179795508, −7.57408163713260395572347788342, −7.42164402326711539791851847806, −6.23404215448878059348217690770, −5.61099808585520850066870601497, −4.64147763449060596166216500116, −4.41767046306567983198621363902, −3.08319609192141911573170938004, −2.47737819392386807936473953446, −1.40466375222577551814353745380, −0.29108879582776425324997284560,
0.29108879582776425324997284560, 1.40466375222577551814353745380, 2.47737819392386807936473953446, 3.08319609192141911573170938004, 4.41767046306567983198621363902, 4.64147763449060596166216500116, 5.61099808585520850066870601497, 6.23404215448878059348217690770, 7.42164402326711539791851847806, 7.57408163713260395572347788342, 8.613331823451940328379179795508, 8.828994686734232356702640091435, 10.06066787452777894299391762156, 10.30370969622166582664606012401, 10.65459078969536679186789192866, 11.56966940314310702646563630120, 12.57518152121079988791880939437, 12.59098893486709497182235246679, 13.35279222730060506210538647871, 13.68781485271226063105839512204
