L(s) = 1 | − 1.41·3-s + 5-s + 1.00·9-s + 1.41·11-s − 13-s − 1.41·15-s − 1.41·19-s + 1.41·23-s + 25-s + 1.41·31-s − 2.00·33-s + 1.41·39-s + 1.41·43-s + 1.00·45-s + 49-s + 1.41·55-s + 2.00·57-s − 1.41·59-s − 65-s − 2.00·69-s − 1.41·71-s − 1.41·75-s − 0.999·81-s − 2.00·93-s − 1.41·95-s + 2·97-s + 1.41·99-s + ⋯ |
L(s) = 1 | − 1.41·3-s + 5-s + 1.00·9-s + 1.41·11-s − 13-s − 1.41·15-s − 1.41·19-s + 1.41·23-s + 25-s + 1.41·31-s − 2.00·33-s + 1.41·39-s + 1.41·43-s + 1.00·45-s + 49-s + 1.41·55-s + 2.00·57-s − 1.41·59-s − 65-s − 2.00·69-s − 1.41·71-s − 1.41·75-s − 0.999·81-s − 2.00·93-s − 1.41·95-s + 2·97-s + 1.41·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8007188920\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8007188920\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 2T + T^{2} \) |
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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
−10.31765061665574087844485722914, −9.387111622254968892235951536332, −8.753761679550678651105072366182, −7.23928750576835164294130054527, −6.48202796561858240370163076165, −6.02442081775765322030223985336, −5.00518514642467107248504673578, −4.32058639621591506813865591005, −2.60812812836947832485503290295, −1.20148064228717156607798095654,
1.20148064228717156607798095654, 2.60812812836947832485503290295, 4.32058639621591506813865591005, 5.00518514642467107248504673578, 6.02442081775765322030223985336, 6.48202796561858240370163076165, 7.23928750576835164294130054527, 8.753761679550678651105072366182, 9.387111622254968892235951536332, 10.31765061665574087844485722914