| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s + 6·6-s − 5·7-s − 4·8-s + 6·9-s − 9·12-s + 2·13-s + 10·14-s + 5·16-s − 6·17-s − 12·18-s + 4·19-s + 15·21-s + 12·24-s + 7·25-s − 4·26-s − 9·27-s − 15·28-s − 16·31-s − 6·32-s + 12·34-s + 18·36-s − 8·38-s − 6·39-s − 30·42-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s + 2.44·6-s − 1.88·7-s − 1.41·8-s + 2·9-s − 2.59·12-s + 0.554·13-s + 2.67·14-s + 5/4·16-s − 1.45·17-s − 2.82·18-s + 0.917·19-s + 3.27·21-s + 2.44·24-s + 7/5·25-s − 0.784·26-s − 1.73·27-s − 2.83·28-s − 2.87·31-s − 1.06·32-s + 2.05·34-s + 3·36-s − 1.29·38-s − 0.960·39-s − 4.62·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1134861854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1134861854\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.98744514629988858298932280200, −10.47940537339231293610537384530, −10.16811071041008149835523994473, −9.902753679929592513909221530771, −9.184848695926766363294939809326, −8.967462412349409188519111699868, −8.726364576074737566685452204941, −7.69830171852438734904832294612, −7.33273285142611350748768584801, −6.77010883631986819751188178852, −6.62202373195864845150671672461, −6.26677459256349196366371373804, −5.63466075184820213684113710937, −5.21843143127830986396278176720, −4.53724676716970795690871960480, −3.42246719177726680751014629698, −3.41251745273826628381294344249, −2.21039341179798672450867367705, −1.34458609380992478097970620333, −0.28739572070691110036812848507,
0.28739572070691110036812848507, 1.34458609380992478097970620333, 2.21039341179798672450867367705, 3.41251745273826628381294344249, 3.42246719177726680751014629698, 4.53724676716970795690871960480, 5.21843143127830986396278176720, 5.63466075184820213684113710937, 6.26677459256349196366371373804, 6.62202373195864845150671672461, 6.77010883631986819751188178852, 7.33273285142611350748768584801, 7.69830171852438734904832294612, 8.726364576074737566685452204941, 8.967462412349409188519111699868, 9.184848695926766363294939809326, 9.902753679929592513909221530771, 10.16811071041008149835523994473, 10.47940537339231293610537384530, 10.98744514629988858298932280200
