| L(s) = 1 | − 2·3-s + 3·9-s − 8·11-s + 4·17-s − 4·19-s + 6·25-s − 4·27-s + 16·33-s + 16·41-s + 8·43-s − 10·49-s − 8·51-s + 8·57-s − 16·59-s + 4·67-s − 20·73-s − 12·75-s + 5·81-s − 8·89-s − 4·97-s − 24·99-s + 24·107-s + 12·113-s + 26·121-s − 32·123-s + 127-s − 16·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 2.41·11-s + 0.970·17-s − 0.917·19-s + 6/5·25-s − 0.769·27-s + 2.78·33-s + 2.49·41-s + 1.21·43-s − 1.42·49-s − 1.12·51-s + 1.05·57-s − 2.08·59-s + 0.488·67-s − 2.34·73-s − 1.38·75-s + 5/9·81-s − 0.847·89-s − 0.406·97-s − 2.41·99-s + 2.32·107-s + 1.12·113-s + 2.36·121-s − 2.88·123-s + 0.0887·127-s − 1.40·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.227808649057000200498276335540, −7.965645572263157998665195857010, −7.33786979345918290465240439010, −7.29595882657599730453498777162, −6.41649036742480464627329082855, −5.89249065254499898615542236081, −5.72491179107937457974413403419, −5.14896024739962723325634761707, −4.50008355692744503094190412349, −4.45984308910786813852795903758, −3.27214076659366118813611445645, −2.81997084527132682127236815913, −2.12656331686074115056549242086, −1.02919547182981641476270794586, 0,
1.02919547182981641476270794586, 2.12656331686074115056549242086, 2.81997084527132682127236815913, 3.27214076659366118813611445645, 4.45984308910786813852795903758, 4.50008355692744503094190412349, 5.14896024739962723325634761707, 5.72491179107937457974413403419, 5.89249065254499898615542236081, 6.41649036742480464627329082855, 7.29595882657599730453498777162, 7.33786979345918290465240439010, 7.965645572263157998665195857010, 8.227808649057000200498276335540
