| L(s) = 1 | − 2·3-s − 8·5-s + 3·9-s − 4·11-s + 16·15-s + 38·25-s − 4·27-s − 20·31-s + 8·33-s + 20·37-s − 24·45-s − 8·47-s − 10·49-s − 20·53-s + 32·55-s − 16·59-s + 4·67-s + 32·71-s − 76·75-s + 5·81-s − 8·89-s + 40·93-s − 4·97-s − 12·99-s − 16·103-s − 40·111-s + 12·113-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3.57·5-s + 9-s − 1.20·11-s + 4.13·15-s + 38/5·25-s − 0.769·27-s − 3.59·31-s + 1.39·33-s + 3.28·37-s − 3.57·45-s − 1.16·47-s − 1.42·49-s − 2.74·53-s + 4.31·55-s − 2.08·59-s + 0.488·67-s + 3.79·71-s − 8.77·75-s + 5/9·81-s − 0.847·89-s + 4.14·93-s − 0.406·97-s − 1.20·99-s − 1.57·103-s − 3.79·111-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−7.33786979345918290465240439010, −6.79221857949224094747221762876, −6.49034317909005458001649653668, −5.89249065254499898615542236081, −5.15899682105998916020946839324, −5.04672662996870790468388922972, −4.45984308910786813852795903758, −4.22935181861929498130806818170, −3.70197408877131997021875612692, −3.27214076659366118813611445645, −2.94520094338462098028115569880, −1.85712821666465876352023788287, −0.863922602362420866192062319772, 0, 0,
0.863922602362420866192062319772, 1.85712821666465876352023788287, 2.94520094338462098028115569880, 3.27214076659366118813611445645, 3.70197408877131997021875612692, 4.22935181861929498130806818170, 4.45984308910786813852795903758, 5.04672662996870790468388922972, 5.15899682105998916020946839324, 5.89249065254499898615542236081, 6.49034317909005458001649653668, 6.79221857949224094747221762876, 7.33786979345918290465240439010
