| L(s) = 1 | − 6·3-s + 21·9-s − 4·13-s − 10·17-s − 12·23-s + 25-s − 54·27-s − 8·29-s + 24·39-s − 6·43-s + 13·49-s + 60·51-s − 4·53-s − 24·61-s + 72·69-s − 6·75-s + 12·79-s + 108·81-s + 48·87-s − 8·101-s − 12·103-s + 24·107-s − 28·113-s − 84·117-s + 6·121-s + 127-s + 36·129-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 7·9-s − 1.10·13-s − 2.42·17-s − 2.50·23-s + 1/5·25-s − 10.3·27-s − 1.48·29-s + 3.84·39-s − 0.914·43-s + 13/7·49-s + 8.40·51-s − 0.549·53-s − 3.07·61-s + 8.66·69-s − 0.692·75-s + 1.35·79-s + 12·81-s + 5.14·87-s − 0.796·101-s − 1.18·103-s + 2.32·107-s − 2.63·113-s − 7.76·117-s + 6/11·121-s + 0.0887·127-s + 3.16·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−10.90058415489492309293663788560, −10.72301841796029202465661168657, −10.45065652592070170228992426050, −9.867705803055531930194665540861, −9.428483445700316367073144700927, −8.924066083756116351624652023821, −7.944595749674271891760273873132, −7.46646046416840431551234300796, −7.00599394138295271063141268226, −6.47010685473241215068343474096, −6.20214531995203480907610222723, −5.83759866425758622327451266892, −5.10019492592589482081221636138, −4.96700719456017829763534961263, −4.12187527325456957596345917235, −4.11737137647606119361774060872, −2.31848070166838050919277286340, −1.58958060048972467946282313599, 0, 0,
1.58958060048972467946282313599, 2.31848070166838050919277286340, 4.11737137647606119361774060872, 4.12187527325456957596345917235, 4.96700719456017829763534961263, 5.10019492592589482081221636138, 5.83759866425758622327451266892, 6.20214531995203480907610222723, 6.47010685473241215068343474096, 7.00599394138295271063141268226, 7.46646046416840431551234300796, 7.944595749674271891760273873132, 8.924066083756116351624652023821, 9.428483445700316367073144700927, 9.867705803055531930194665540861, 10.45065652592070170228992426050, 10.72301841796029202465661168657, 10.90058415489492309293663788560
