| L(s) = 1 | − 2·5-s + 2·7-s + 4·13-s − 6·17-s + 2·19-s + 3·25-s − 12·29-s + 2·31-s − 4·35-s − 2·37-s − 12·41-s − 10·43-s + 6·47-s − 8·49-s − 18·53-s + 12·59-s − 8·61-s − 8·65-s + 8·67-s + 12·71-s + 10·73-s + 8·79-s + 6·83-s + 12·85-s + 8·91-s − 4·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.10·13-s − 1.45·17-s + 0.458·19-s + 3/5·25-s − 2.22·29-s + 0.359·31-s − 0.676·35-s − 0.328·37-s − 1.87·41-s − 1.52·43-s + 0.875·47-s − 8/7·49-s − 2.47·53-s + 1.56·59-s − 1.02·61-s − 0.992·65-s + 0.977·67-s + 1.42·71-s + 1.17·73-s + 0.900·79-s + 0.658·83-s + 1.30·85-s + 0.838·91-s − 0.410·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 144 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 192 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.83009491592755100131526506928, −7.75659896285194814676967293204, −7.02403237266723271975213801864, −6.74262447421080107363240017795, −6.54838572286417753928769378025, −6.28703012698476570934151608259, −5.48598913264094180704029568188, −5.34597140177740846120935457255, −4.93667746430803761233177687816, −4.70074375827062317320118910551, −4.04737186974690732384457020552, −3.84201589006191295576647674748, −3.45990487785916009664328650431, −3.20950967664682208863890373998, −2.38446523144177306074597359986, −2.10421258721886091231040919431, −1.43417220710363950047592495381, −1.24195182677413151914631040688, 0, 0,
1.24195182677413151914631040688, 1.43417220710363950047592495381, 2.10421258721886091231040919431, 2.38446523144177306074597359986, 3.20950967664682208863890373998, 3.45990487785916009664328650431, 3.84201589006191295576647674748, 4.04737186974690732384457020552, 4.70074375827062317320118910551, 4.93667746430803761233177687816, 5.34597140177740846120935457255, 5.48598913264094180704029568188, 6.28703012698476570934151608259, 6.54838572286417753928769378025, 6.74262447421080107363240017795, 7.02403237266723271975213801864, 7.75659896285194814676967293204, 7.83009491592755100131526506928
