L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s − 6·13-s + 16-s − 4·17-s − 18-s − 6·19-s + 22-s − 4·23-s + 24-s − 5·25-s + 6·26-s − 27-s + 6·29-s + 2·31-s − 32-s + 33-s + 4·34-s + 36-s + 10·37-s + 6·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.37·19-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 25-s + 1.17·26-s − 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s + 0.685·34-s + 1/6·36-s + 1.64·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5572233064\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5572233064\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.652552034480864194119013053515, −7.83922398202642458587198828500, −7.30932067156140291452614810498, −6.37119763678498512356952453412, −5.90660738221292552490908354487, −4.70213202876017617150319733849, −4.23003763111100218531402281021, −2.63621530561166833588250288036, −2.07103752121128576013331888677, −0.48386265623252023695541121788,
0.48386265623252023695541121788, 2.07103752121128576013331888677, 2.63621530561166833588250288036, 4.23003763111100218531402281021, 4.70213202876017617150319733849, 5.90660738221292552490908354487, 6.37119763678498512356952453412, 7.30932067156140291452614810498, 7.83922398202642458587198828500, 8.652552034480864194119013053515