| L(s) = 1 | + 3-s + 5-s + 2·11-s − 8·13-s + 15-s − 2·17-s − 4·19-s − 6·23-s − 4·25-s + 2·27-s − 5·31-s + 2·33-s + 2·37-s − 8·39-s − 7·41-s − 43-s − 6·47-s − 2·51-s − 15·53-s + 2·55-s − 4·57-s − 14·59-s − 9·61-s − 8·65-s + 67-s − 6·69-s + 18·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.603·11-s − 2.21·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s − 1.25·23-s − 4/5·25-s + 0.384·27-s − 0.898·31-s + 0.348·33-s + 0.328·37-s − 1.28·39-s − 1.09·41-s − 0.152·43-s − 0.875·47-s − 0.280·51-s − 2.06·53-s + 0.269·55-s − 0.529·57-s − 1.82·59-s − 1.15·61-s − 0.992·65-s + 0.122·67-s − 0.722·69-s + 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 89 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 81 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 157 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 146 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 137 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 202 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 25 T + 297 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 170 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 245 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.394574882230178462323055196187, −8.144004723044429949681000474943, −7.65020941506743886520235105228, −7.25646756782276450804835106539, −7.22476923422507437284312042975, −6.44732848220904090216194438674, −6.12174373467520294179781685223, −6.08139568799959490954076140707, −5.32963711216213509309440371141, −4.83571668208539413252543658774, −4.60575524283424273019311188737, −4.30935336920585011045844191071, −3.60813392503252034933952291889, −3.26262521549816939732752253526, −2.72671715483622065217592137333, −2.25486934919605927145213512251, −1.83753287166769564319053053887, −1.51587604481610163478791928692, 0, 0,
1.51587604481610163478791928692, 1.83753287166769564319053053887, 2.25486934919605927145213512251, 2.72671715483622065217592137333, 3.26262521549816939732752253526, 3.60813392503252034933952291889, 4.30935336920585011045844191071, 4.60575524283424273019311188737, 4.83571668208539413252543658774, 5.32963711216213509309440371141, 6.08139568799959490954076140707, 6.12174373467520294179781685223, 6.44732848220904090216194438674, 7.22476923422507437284312042975, 7.25646756782276450804835106539, 7.65020941506743886520235105228, 8.144004723044429949681000474943, 8.394574882230178462323055196187
