| L(s) = 1 | + 2·5-s + 2·11-s + 2·13-s − 8·17-s + 2·19-s + 2·23-s + 3·25-s − 12·29-s − 10·31-s − 8·41-s + 18·43-s + 16·47-s − 2·49-s + 8·53-s + 4·55-s − 6·59-s − 4·61-s + 4·65-s + 12·67-s + 22·71-s + 8·73-s + 20·79-s − 16·85-s + 12·89-s + 4·95-s + 4·97-s + 4·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.603·11-s + 0.554·13-s − 1.94·17-s + 0.458·19-s + 0.417·23-s + 3/5·25-s − 2.22·29-s − 1.79·31-s − 1.24·41-s + 2.74·43-s + 2.33·47-s − 2/7·49-s + 1.09·53-s + 0.539·55-s − 0.781·59-s − 0.512·61-s + 0.496·65-s + 1.46·67-s + 2.61·71-s + 0.936·73-s + 2.25·79-s − 1.73·85-s + 1.27·89-s + 0.410·95-s + 0.406·97-s + 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.337862951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.337862951\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 18 T + 164 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22 T + 260 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.66932797472567345400132663529, −7.50227907751273335090933801862, −7.15881734026001285415719230248, −6.93822566600612349458449296456, −6.30768282721535510239365128859, −6.29311803748756998904354046834, −5.90444578398950514225597736986, −5.42890472133352960492576215587, −5.16333783735678506011554086416, −4.97129073459387714707069561790, −4.22752909319913215474758985264, −4.02446855031379907917212036784, −3.55041871822308139440250505164, −3.53672837713479677734697002205, −2.57811133880665419628749329243, −2.37032137546194039552553270966, −1.82979545246846303763571298677, −1.82818494036223314455113471899, −0.820355759678632141622631165770, −0.58370147085212629911437810129,
0.58370147085212629911437810129, 0.820355759678632141622631165770, 1.82818494036223314455113471899, 1.82979545246846303763571298677, 2.37032137546194039552553270966, 2.57811133880665419628749329243, 3.53672837713479677734697002205, 3.55041871822308139440250505164, 4.02446855031379907917212036784, 4.22752909319913215474758985264, 4.97129073459387714707069561790, 5.16333783735678506011554086416, 5.42890472133352960492576215587, 5.90444578398950514225597736986, 6.29311803748756998904354046834, 6.30768282721535510239365128859, 6.93822566600612349458449296456, 7.15881734026001285415719230248, 7.50227907751273335090933801862, 7.66932797472567345400132663529
