| L(s) = 1 | + 3·2-s + 3·3-s + 4·4-s + 9·6-s + 4·7-s + 3·8-s + 2·9-s − 7·11-s + 12·12-s + 10·13-s + 12·14-s + 3·16-s + 12·17-s + 6·18-s + 12·21-s − 21·22-s + 23-s + 9·24-s + 30·26-s − 6·27-s + 16·28-s − 3·29-s − 15·31-s + 6·32-s − 21·33-s + 36·34-s + 8·36-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 2·4-s + 3.67·6-s + 1.51·7-s + 1.06·8-s + 2/3·9-s − 2.11·11-s + 3.46·12-s + 2.77·13-s + 3.20·14-s + 3/4·16-s + 2.91·17-s + 1.41·18-s + 2.61·21-s − 4.47·22-s + 0.208·23-s + 1.83·24-s + 5.88·26-s − 1.15·27-s + 3.02·28-s − 0.557·29-s − 2.69·31-s + 1.06·32-s − 3.65·33-s + 6.17·34-s + 4/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(30.10822947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(30.10822947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 3 p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 15 T + 107 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 153 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 125 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 117 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 173 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 133 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 163 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−7.912127779681261022350646881487, −7.52851597745067012379326165114, −7.37324950997681724327742219397, −7.14273218844496553454512672900, −6.09370336666344781699456935291, −5.87976227655987939306422626546, −5.60386328184930433799949845770, −5.57001208124548262802425304571, −5.14323298063456657022459261880, −4.89047327111997569236784197042, −4.11055062214229240704337684736, −3.99073769033421495770545300200, −3.66754665522425879512164238349, −3.35898415791828774566415791333, −3.05192897508004427481270958147, −2.73615561979406006171787491749, −2.24261716420937334428272546686, −1.68113756773508305348519639633, −1.39732465934610180722098323874, −0.73757425667811069209212819860,
0.73757425667811069209212819860, 1.39732465934610180722098323874, 1.68113756773508305348519639633, 2.24261716420937334428272546686, 2.73615561979406006171787491749, 3.05192897508004427481270958147, 3.35898415791828774566415791333, 3.66754665522425879512164238349, 3.99073769033421495770545300200, 4.11055062214229240704337684736, 4.89047327111997569236784197042, 5.14323298063456657022459261880, 5.57001208124548262802425304571, 5.60386328184930433799949845770, 5.87976227655987939306422626546, 6.09370336666344781699456935291, 7.14273218844496553454512672900, 7.37324950997681724327742219397, 7.52851597745067012379326165114, 7.912127779681261022350646881487
