L(s) = 1 | − 3-s − 5-s + 9-s − 0.828·11-s + 15-s − 2.82·17-s + 1.41·19-s + 3.41·23-s + 25-s − 27-s − 2·29-s − 5.41·31-s + 0.828·33-s + 3.17·37-s − 0.343·41-s + 5.65·43-s − 45-s − 0.828·47-s + 2.82·51-s + 0.585·53-s + 0.828·55-s − 1.41·57-s − 8.48·59-s + 7.07·61-s + 6.82·67-s − 3.41·69-s + 7.65·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.333·9-s − 0.249·11-s + 0.258·15-s − 0.685·17-s + 0.324·19-s + 0.711·23-s + 0.200·25-s − 0.192·27-s − 0.371·29-s − 0.972·31-s + 0.144·33-s + 0.521·37-s − 0.0535·41-s + 0.862·43-s − 0.149·45-s − 0.120·47-s + 0.396·51-s + 0.0804·53-s + 0.111·55-s − 0.187·57-s − 1.10·59-s + 0.905·61-s + 0.834·67-s − 0.411·69-s + 0.908·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.128339682\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128339682\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 + 0.343T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 - 0.585T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 - 7.65T + 71T^{2} \) |
| 73 | \( 1 + 7.31T + 73T^{2} \) |
| 79 | \( 1 - 0.343T + 79T^{2} \) |
| 83 | \( 1 - 4.82T + 83T^{2} \) |
| 89 | \( 1 - 6.48T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.821040589355091632675547167070, −7.85923732410970340716533853071, −7.25395923441861178816486182192, −6.50830382956679696245206676075, −5.64620129080780264887971573847, −4.90601854856824341232407880038, −4.10979787120273572081673832786, −3.17720454925745162031509051072, −2.01759213698838676981285743291, −0.66224298514062171322900423013,
0.66224298514062171322900423013, 2.01759213698838676981285743291, 3.17720454925745162031509051072, 4.10979787120273572081673832786, 4.90601854856824341232407880038, 5.64620129080780264887971573847, 6.50830382956679696245206676075, 7.25395923441861178816486182192, 7.85923732410970340716533853071, 8.821040589355091632675547167070