L(s) = 1 | − 4·11-s − 6·13-s − 6·17-s + 4·19-s + 2·29-s + 8·31-s + 2·37-s + 6·41-s + 12·43-s − 8·47-s − 7·49-s + 6·53-s + 12·59-s + 14·61-s + 4·67-s + 8·71-s + 6·73-s + 8·79-s + 12·83-s − 10·89-s − 2·97-s − 6·101-s + 4·107-s − 18·109-s − 6·113-s + ⋯ |
L(s) = 1 | − 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s + 0.371·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s + 1.82·43-s − 1.16·47-s − 49-s + 0.824·53-s + 1.56·59-s + 1.79·61-s + 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.900·79-s + 1.31·83-s − 1.05·89-s − 0.203·97-s − 0.597·101-s + 0.386·107-s − 1.72·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.311128052\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311128052\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.391564530969274177614715751500, −7.84433918506098122901785888900, −7.10648850833945378873095434686, −6.43925783086057872365275625126, −5.31059979041689708302017443740, −4.90112278095406188180879238277, −4.01130829705839256794389128825, −2.63672797798552915147078140108, −2.38987395166756440433579405452, −0.63962793409840065289934962638,
0.63962793409840065289934962638, 2.38987395166756440433579405452, 2.63672797798552915147078140108, 4.01130829705839256794389128825, 4.90112278095406188180879238277, 5.31059979041689708302017443740, 6.43925783086057872365275625126, 7.10648850833945378873095434686, 7.84433918506098122901785888900, 8.391564530969274177614715751500