L(s) = 1 | − 4·7-s + 6·13-s − 2·17-s + 4·19-s − 8·23-s + 6·29-s + 6·37-s − 10·41-s + 4·43-s + 8·47-s + 9·49-s + 10·53-s + 6·61-s + 4·67-s + 14·73-s + 16·79-s + 12·83-s − 2·89-s − 24·91-s − 2·97-s + 14·101-s − 4·103-s + 4·107-s − 10·109-s + 6·113-s + 8·119-s + ⋯ |
L(s) = 1 | − 1.51·7-s + 1.66·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1.11·29-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s + 0.768·61-s + 0.488·67-s + 1.63·73-s + 1.80·79-s + 1.31·83-s − 0.211·89-s − 2.51·91-s − 0.203·97-s + 1.39·101-s − 0.394·103-s + 0.386·107-s − 0.957·109-s + 0.564·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.444187553\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444187553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.292583113211074106889707915361, −8.571128295571040401608112310840, −7.75290550331790987147863139637, −6.59528886370780842701990768227, −6.28693232043460858847346479420, −5.38747455398254339407810680866, −4.00473327983531279447179237561, −3.48877015763051580885745741856, −2.37946394700543657872848619881, −0.819406966412527131065932234969,
0.819406966412527131065932234969, 2.37946394700543657872848619881, 3.48877015763051580885745741856, 4.00473327983531279447179237561, 5.38747455398254339407810680866, 6.28693232043460858847346479420, 6.59528886370780842701990768227, 7.75290550331790987147863139637, 8.571128295571040401608112310840, 9.292583113211074106889707915361