L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 11-s + 7·13-s + 16-s + 2·17-s + 2·20-s − 22-s + 8·23-s − 25-s − 7·26-s + 5·29-s − 4·31-s − 32-s − 2·34-s + 4·37-s − 2·40-s + 4·41-s − 8·43-s + 44-s − 8·46-s + 2·47-s + 50-s + 7·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 0.301·11-s + 1.94·13-s + 1/4·16-s + 0.485·17-s + 0.447·20-s − 0.213·22-s + 1.66·23-s − 1/5·25-s − 1.37·26-s + 0.928·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s + 0.657·37-s − 0.316·40-s + 0.624·41-s − 1.21·43-s + 0.150·44-s − 1.17·46-s + 0.291·47-s + 0.141·50-s + 0.970·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.385817641\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.385817641\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.84477951969434096310577982884, −6.83219647419296523001917030474, −6.48124991250803522400862921820, −5.73463467816586580011825936375, −5.19411244710029851212503471851, −4.03129780667914228339613638582, −3.31941568253155764991923042646, −2.46365880421015085487796170449, −1.45683528128934257946421228554, −0.930433631277048359621158016997,
0.930433631277048359621158016997, 1.45683528128934257946421228554, 2.46365880421015085487796170449, 3.31941568253155764991923042646, 4.03129780667914228339613638582, 5.19411244710029851212503471851, 5.73463467816586580011825936375, 6.48124991250803522400862921820, 6.83219647419296523001917030474, 7.84477951969434096310577982884