L(s) = 1 | − 2-s − 0.732·3-s + 4-s − 5-s + 0.732·6-s − 7-s − 8-s − 2.46·9-s + 10-s − 0.732·12-s − 5.46·13-s + 14-s + 0.732·15-s + 16-s − 3.46·17-s + 2.46·18-s − 3.26·19-s − 20-s + 0.732·21-s + 2.19·23-s + 0.732·24-s + 25-s + 5.46·26-s + 4·27-s − 28-s + 1.26·29-s − 0.732·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.422·3-s + 0.5·4-s − 0.447·5-s + 0.298·6-s − 0.377·7-s − 0.353·8-s − 0.821·9-s + 0.316·10-s − 0.211·12-s − 1.51·13-s + 0.267·14-s + 0.189·15-s + 0.250·16-s − 0.840·17-s + 0.580·18-s − 0.749·19-s − 0.223·20-s + 0.159·21-s + 0.457·23-s + 0.149·24-s + 0.200·25-s + 1.07·26-s + 0.769·27-s − 0.188·28-s + 0.235·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.07326858123\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07326858123\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.962022869841743540773668008923, −6.91293904617056177845845610519, −6.71941525247226196483818943579, −5.84157406733271623585095954346, −4.97904534958620749140701463199, −4.43723244118109288391690677305, −3.19437633866738182789122358667, −2.67997533242755250837786509196, −1.65609134455063738807448029125, −0.14404834769734862854217692458,
0.14404834769734862854217692458, 1.65609134455063738807448029125, 2.67997533242755250837786509196, 3.19437633866738182789122358667, 4.43723244118109288391690677305, 4.97904534958620749140701463199, 5.84157406733271623585095954346, 6.71941525247226196483818943579, 6.91293904617056177845845610519, 7.962022869841743540773668008923