| L(s) = 1 | + 2·3-s − 2·5-s − 9-s − 2·11-s + 10·13-s − 4·15-s + 10·17-s − 4·19-s + 4·23-s + 3·25-s − 6·27-s + 2·29-s + 12·31-s − 4·33-s − 4·37-s + 20·39-s + 4·41-s + 12·43-s + 2·45-s + 2·47-s + 20·51-s − 16·53-s + 4·55-s − 8·57-s + 4·59-s + 16·61-s − 20·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1/3·9-s − 0.603·11-s + 2.77·13-s − 1.03·15-s + 2.42·17-s − 0.917·19-s + 0.834·23-s + 3/5·25-s − 1.15·27-s + 0.371·29-s + 2.15·31-s − 0.696·33-s − 0.657·37-s + 3.20·39-s + 0.624·41-s + 1.82·43-s + 0.298·45-s + 0.291·47-s + 2.80·51-s − 2.19·53-s + 0.539·55-s − 1.05·57-s + 0.520·59-s + 2.04·61-s − 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.182079646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.182079646\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 T + 49 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 152 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 100 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.18181289337866264006377234543, −9.816119784017524976327792242485, −9.104550421850935656851454936345, −8.831624190923958194825477737784, −8.523735596558068509988176447842, −8.125291161511137798438079282916, −7.77418011814709207573770323301, −7.75875209760885542833696795654, −6.72810602975894477120995168291, −6.49222971907221780121271830978, −5.82417046577116847520388913109, −5.61668618531821323409870696066, −4.92618062324354990304610105151, −4.24294711361137898487129725041, −3.69050598221504642462385922343, −3.52273962006566450914630672286, −2.77078799651627360286253031598, −2.70352993732482987557137195511, −1.36532043253345373940759128199, −0.894208462432729505905730397256,
0.894208462432729505905730397256, 1.36532043253345373940759128199, 2.70352993732482987557137195511, 2.77078799651627360286253031598, 3.52273962006566450914630672286, 3.69050598221504642462385922343, 4.24294711361137898487129725041, 4.92618062324354990304610105151, 5.61668618531821323409870696066, 5.82417046577116847520388913109, 6.49222971907221780121271830978, 6.72810602975894477120995168291, 7.75875209760885542833696795654, 7.77418011814709207573770323301, 8.125291161511137798438079282916, 8.523735596558068509988176447842, 8.831624190923958194825477737784, 9.104550421850935656851454936345, 9.816119784017524976327792242485, 10.18181289337866264006377234543
