L(s) = 1 | − 3-s + 9-s − 4·11-s + 13-s + 8·17-s + 5·19-s − 4·23-s − 27-s + 9·29-s − 4·31-s + 4·33-s − 3·37-s − 39-s + 5·41-s + 6·43-s − 5·47-s − 7·49-s − 8·51-s − 5·53-s − 5·57-s − 6·59-s + 4·61-s − 3·67-s + 4·69-s + 7·71-s + 4·73-s − 79-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.94·17-s + 1.14·19-s − 0.834·23-s − 0.192·27-s + 1.67·29-s − 0.718·31-s + 0.696·33-s − 0.493·37-s − 0.160·39-s + 0.780·41-s + 0.914·43-s − 0.729·47-s − 49-s − 1.12·51-s − 0.686·53-s − 0.662·57-s − 0.781·59-s + 0.512·61-s − 0.366·67-s + 0.481·69-s + 0.830·71-s + 0.468·73-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.560895097\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.560895097\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.894581814392885404863826825593, −7.25467116205847708699360954926, −6.36788578736367735009577181895, −5.61021863169485937156407084434, −5.26990243176438988878830460320, −4.43686154150801565464605012312, −3.40925856449040420640788274159, −2.83753397807738956315745825713, −1.60408405829712347246377767977, −0.66606564550893745249851846224,
0.66606564550893745249851846224, 1.60408405829712347246377767977, 2.83753397807738956315745825713, 3.40925856449040420640788274159, 4.43686154150801565464605012312, 5.26990243176438988878830460320, 5.61021863169485937156407084434, 6.36788578736367735009577181895, 7.25467116205847708699360954926, 7.894581814392885404863826825593