| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s − 4·11-s + 6·12-s + 5·16-s − 8·17-s − 6·18-s − 2·19-s + 8·22-s − 4·23-s − 8·24-s + 2·25-s + 4·27-s − 12·29-s − 6·32-s − 8·33-s + 16·34-s + 9·36-s + 20·37-s + 4·38-s + 4·41-s − 12·44-s + 8·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s − 1.20·11-s + 1.73·12-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 0.458·19-s + 1.70·22-s − 0.834·23-s − 1.63·24-s + 2/5·25-s + 0.769·27-s − 2.22·29-s − 1.06·32-s − 1.39·33-s + 2.74·34-s + 3/2·36-s + 3.28·37-s + 0.648·38-s + 0.624·41-s − 1.80·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 246 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.992623530692950454339934098052, −7.81527249228281624034637417497, −7.37640818060576654741324481379, −7.16433466454858379842460664965, −6.73312677699122140700257445786, −6.33209536284402095642074737346, −5.83520585735170778549203290538, −5.75180339170024169886293020272, −5.04440351078505680859734675380, −4.49164835732825445009880234514, −4.14574308441539835543886498604, −3.99046128636283841912109211178, −3.11307018219718537892976293007, −2.84301183057104393501007275263, −2.45238323642217898084925885612, −2.16733463971007221010120316165, −1.66890000642658293643010932139, −1.14197829333841597024086891974, 0, 0,
1.14197829333841597024086891974, 1.66890000642658293643010932139, 2.16733463971007221010120316165, 2.45238323642217898084925885612, 2.84301183057104393501007275263, 3.11307018219718537892976293007, 3.99046128636283841912109211178, 4.14574308441539835543886498604, 4.49164835732825445009880234514, 5.04440351078505680859734675380, 5.75180339170024169886293020272, 5.83520585735170778549203290538, 6.33209536284402095642074737346, 6.73312677699122140700257445786, 7.16433466454858379842460664965, 7.37640818060576654741324481379, 7.81527249228281624034637417497, 7.992623530692950454339934098052
