| L(s) = 1 | + 2·5-s + 6·7-s − 2·11-s + 6·13-s + 2·19-s + 4·23-s + 3·25-s − 6·29-s + 12·35-s + 2·37-s + 6·41-s + 6·43-s + 4·47-s + 18·49-s − 8·53-s − 4·55-s + 12·59-s + 12·65-s + 12·67-s + 16·73-s − 12·77-s − 8·83-s − 10·89-s + 36·91-s + 4·95-s − 14·97-s + 28·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.26·7-s − 0.603·11-s + 1.66·13-s + 0.458·19-s + 0.834·23-s + 3/5·25-s − 1.11·29-s + 2.02·35-s + 0.328·37-s + 0.937·41-s + 0.914·43-s + 0.583·47-s + 18/7·49-s − 1.09·53-s − 0.539·55-s + 1.56·59-s + 1.48·65-s + 1.46·67-s + 1.87·73-s − 1.36·77-s − 0.878·83-s − 1.05·89-s + 3.77·91-s + 0.410·95-s − 1.42·97-s + 2.75·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46785600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46785600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.825212944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.825212944\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.124356408694151834427347085869, −7.87990044780814652222378503137, −7.44690062104442250101359635298, −7.28533704989434249056731873913, −6.56366173577286990451969727169, −6.53612594300121925932649227890, −5.73855621667658530456705381659, −5.71511838170767434762902238401, −5.27721586424447143641806594096, −5.17005435552545685518930338771, −4.46195137849900134905962177317, −4.38084393641928110390797943220, −3.82515413829852694352333434621, −3.40925684941585861492184765343, −2.85981373633926266754249138638, −2.38412167292024674129098985164, −1.89776623332570026488386716960, −1.70443795352673202194346302434, −0.939480966635311620287737337731, −0.874037518062871659142406326712,
0.874037518062871659142406326712, 0.939480966635311620287737337731, 1.70443795352673202194346302434, 1.89776623332570026488386716960, 2.38412167292024674129098985164, 2.85981373633926266754249138638, 3.40925684941585861492184765343, 3.82515413829852694352333434621, 4.38084393641928110390797943220, 4.46195137849900134905962177317, 5.17005435552545685518930338771, 5.27721586424447143641806594096, 5.71511838170767434762902238401, 5.73855621667658530456705381659, 6.53612594300121925932649227890, 6.56366173577286990451969727169, 7.28533704989434249056731873913, 7.44690062104442250101359635298, 7.87990044780814652222378503137, 8.124356408694151834427347085869
