| L(s) = 1 | − 2-s − 2.48·3-s + 4-s + 3.25·5-s + 2.48·6-s − 4.58·7-s − 8-s + 3.19·9-s − 3.25·10-s − 2.48·12-s + 4.82·13-s + 4.58·14-s − 8.10·15-s + 16-s − 1.77·17-s − 3.19·18-s + 19-s + 3.25·20-s + 11.4·21-s + 6.51·23-s + 2.48·24-s + 5.62·25-s − 4.82·26-s − 0.475·27-s − 4.58·28-s + 0.0103·29-s + 8.10·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.43·3-s + 0.5·4-s + 1.45·5-s + 1.01·6-s − 1.73·7-s − 0.353·8-s + 1.06·9-s − 1.03·10-s − 0.718·12-s + 1.33·13-s + 1.22·14-s − 2.09·15-s + 0.250·16-s − 0.431·17-s − 0.752·18-s + 0.229·19-s + 0.728·20-s + 2.49·21-s + 1.35·23-s + 0.507·24-s + 1.12·25-s − 0.945·26-s − 0.0915·27-s − 0.867·28-s + 0.00192·29-s + 1.48·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7978667263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7978667263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.48T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 + 4.58T + 7T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 23 | \( 1 - 6.51T + 23T^{2} \) |
| 29 | \( 1 - 0.0103T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 - 0.688T + 37T^{2} \) |
| 41 | \( 1 + 3.27T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 + 6.30T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 + 9.20T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 5.08T + 67T^{2} \) |
| 71 | \( 1 - 9.43T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 2.32T + 83T^{2} \) |
| 89 | \( 1 + 6.04T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−8.692963304079610666990700170568, −7.21645634001423206550266589213, −6.62693869707007691615316101436, −6.19176021642408897373159060168, −5.75070881921090996054398514200, −4.99201288285581401570984465906, −3.62246653082996854545066638988, −2.76988186848826621160177872956, −1.56530370169778899722642609969, −0.61605739798179313399403433901,
0.61605739798179313399403433901, 1.56530370169778899722642609969, 2.76988186848826621160177872956, 3.62246653082996854545066638988, 4.99201288285581401570984465906, 5.75070881921090996054398514200, 6.19176021642408897373159060168, 6.62693869707007691615316101436, 7.21645634001423206550266589213, 8.692963304079610666990700170568
