| L(s) = 1 | − 3-s − 5-s + 2·7-s − 4·9-s − 5·11-s − 13-s + 15-s − 2·21-s + 23-s + 2·25-s + 6·27-s + 3·29-s − 17·31-s + 5·33-s − 2·35-s + 2·37-s + 39-s + 17·41-s − 6·43-s + 4·45-s − 2·47-s − 6·49-s + 8·53-s + 5·55-s − 14·59-s − 19·61-s − 8·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s − 4/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.436·21-s + 0.208·23-s + 2/5·25-s + 1.15·27-s + 0.557·29-s − 3.05·31-s + 0.870·33-s − 0.338·35-s + 0.328·37-s + 0.160·39-s + 2.65·41-s − 0.914·43-s + 0.596·45-s − 0.291·47-s − 6/7·49-s + 1.09·53-s + 0.674·55-s − 1.82·59-s − 2.43·61-s − 1.00·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 3 T - T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 17 T + 133 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 19 T + 211 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 123 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 117 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.769864460941078009362586251491, −8.321906214625450307925073386963, −7.938719086852026653062089827406, −7.76037241063182108831191156877, −7.18371038650827330023163554764, −7.14141541810912673276885485048, −6.25118871682023166550125613831, −5.93413699195548337040859199668, −5.66066900696456155223540626054, −5.26444465352941903649710666135, −4.79349275179450951857527691565, −4.63340670909118477644036268773, −3.93487574639646458794843975015, −3.37898508990300207671263441878, −2.81358716855355318974090931267, −2.62835604648112675911862417268, −1.88273745425703762303955273291, −1.20111670755529859367750701857, 0, 0,
1.20111670755529859367750701857, 1.88273745425703762303955273291, 2.62835604648112675911862417268, 2.81358716855355318974090931267, 3.37898508990300207671263441878, 3.93487574639646458794843975015, 4.63340670909118477644036268773, 4.79349275179450951857527691565, 5.26444465352941903649710666135, 5.66066900696456155223540626054, 5.93413699195548337040859199668, 6.25118871682023166550125613831, 7.14141541810912673276885485048, 7.18371038650827330023163554764, 7.76037241063182108831191156877, 7.938719086852026653062089827406, 8.321906214625450307925073386963, 8.769864460941078009362586251491
