L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 24·7-s − 8·8-s − 10·10-s − 32·11-s + 2·13-s + 48·14-s + 16·16-s − 106·17-s − 19·19-s + 20·20-s + 64·22-s − 152·23-s + 25·25-s − 4·26-s − 96·28-s − 90·29-s + 52·31-s − 32·32-s + 212·34-s − 120·35-s + 306·37-s + 38·38-s − 40·40-s − 62·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.29·7-s − 0.353·8-s − 0.316·10-s − 0.877·11-s + 0.0426·13-s + 0.916·14-s + 1/4·16-s − 1.51·17-s − 0.229·19-s + 0.223·20-s + 0.620·22-s − 1.37·23-s + 1/5·25-s − 0.0301·26-s − 0.647·28-s − 0.576·29-s + 0.301·31-s − 0.176·32-s + 1.06·34-s − 0.579·35-s + 1.35·37-s + 0.162·38-s − 0.158·40-s − 0.236·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5273046789\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5273046789\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 + p T \) |
good | 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 106 T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 52 T + p^{3} T^{2} \) |
| 37 | \( 1 - 306 T + p^{3} T^{2} \) |
| 41 | \( 1 + 62 T + p^{3} T^{2} \) |
| 43 | \( 1 + 268 T + p^{3} T^{2} \) |
| 47 | \( 1 + 456 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 300 T + p^{3} T^{2} \) |
| 61 | \( 1 - 502 T + p^{3} T^{2} \) |
| 67 | \( 1 + 644 T + p^{3} T^{2} \) |
| 71 | \( 1 - 608 T + p^{3} T^{2} \) |
| 73 | \( 1 + 198 T + p^{3} T^{2} \) |
| 79 | \( 1 - 260 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1248 T + p^{3} T^{2} \) |
| 89 | \( 1 + 110 T + p^{3} T^{2} \) |
| 97 | \( 1 + 574 T + p^{3} T^{2} \) |
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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
−9.052274315484558281870937635480, −8.291101169711890054004272740922, −7.44038431303817979881594402154, −6.43305625582112608754777824505, −6.15571774514570208788075462851, −4.95224949597489704741659247441, −3.76926218380155516096984227106, −2.70569402617748958044089463171, −1.96059652869516783373282177230, −0.35902937402663873825912277815,
0.35902937402663873825912277815, 1.96059652869516783373282177230, 2.70569402617748958044089463171, 3.76926218380155516096984227106, 4.95224949597489704741659247441, 6.15571774514570208788075462851, 6.43305625582112608754777824505, 7.44038431303817979881594402154, 8.291101169711890054004272740922, 9.052274315484558281870937635480