L(s) = 1 | + 3-s + 2·5-s + 9-s + 4·11-s + 6·13-s + 2·15-s + 2·17-s + 4·19-s + 4·23-s − 25-s + 27-s − 2·29-s + 8·31-s + 4·33-s − 10·37-s + 6·39-s + 2·41-s − 8·43-s + 2·45-s + 2·51-s − 10·53-s + 8·55-s + 4·57-s − 12·59-s − 10·61-s + 12·65-s + 8·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 1.64·37-s + 0.960·39-s + 0.312·41-s − 1.21·43-s + 0.298·45-s + 0.280·51-s − 1.37·53-s + 1.07·55-s + 0.529·57-s − 1.56·59-s − 1.28·61-s + 1.48·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.730072676\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.730072676\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.418713588968782317564024838754, −7.66327510102367282452091895791, −6.67956200320228921792079181945, −6.25021124460398922393105339706, −5.46186045530175262784183956299, −4.52922956895268217959450215602, −3.53327379023763901445163679786, −3.07868956941990475990299604025, −1.67899038858433256706838035473, −1.23750269317022525830198896132,
1.23750269317022525830198896132, 1.67899038858433256706838035473, 3.07868956941990475990299604025, 3.53327379023763901445163679786, 4.52922956895268217959450215602, 5.46186045530175262784183956299, 6.25021124460398922393105339706, 6.67956200320228921792079181945, 7.66327510102367282452091895791, 8.418713588968782317564024838754