L(s) = 1 | + 3-s + 0.235·5-s + 1.32·7-s + 9-s + 1.55·11-s − 3.66·13-s + 0.235·15-s − 2.58·17-s − 1.45·19-s + 1.32·21-s − 23-s − 4.94·25-s + 27-s + 29-s + 6.25·31-s + 1.55·33-s + 0.311·35-s − 9.15·37-s − 3.66·39-s + 11.0·41-s + 9.13·43-s + 0.235·45-s + 7.96·47-s − 5.25·49-s − 2.58·51-s − 5.80·53-s + 0.367·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.105·5-s + 0.499·7-s + 0.333·9-s + 0.469·11-s − 1.01·13-s + 0.0608·15-s − 0.627·17-s − 0.333·19-s + 0.288·21-s − 0.208·23-s − 0.988·25-s + 0.192·27-s + 0.185·29-s + 1.12·31-s + 0.271·33-s + 0.0526·35-s − 1.50·37-s − 0.587·39-s + 1.73·41-s + 1.39·43-s + 0.0351·45-s + 1.16·47-s − 0.750·49-s − 0.362·51-s − 0.797·53-s + 0.0495·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.634553667\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.634553667\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 5 | \( 1 - 0.235T + 5T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 31 | \( 1 - 6.25T + 31T^{2} \) |
| 37 | \( 1 + 9.15T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 9.13T + 43T^{2} \) |
| 47 | \( 1 - 7.96T + 47T^{2} \) |
| 53 | \( 1 + 5.80T + 53T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 0.0629T + 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 9.93T + 89T^{2} \) |
| 97 | \( 1 + 0.659T + 97T^{2} \) |
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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
−7.922559735532104554314707682007, −7.20726031842460544111307676429, −6.56524989796469319329588569619, −5.76563673267363604751899689414, −4.89115725391632747879882280722, −4.28373735048288416703313255205, −3.57626664076770760413659262096, −2.42939398855383757434466282180, −2.05309755985006742815764141763, −0.77310971636436325791080797292,
0.77310971636436325791080797292, 2.05309755985006742815764141763, 2.42939398855383757434466282180, 3.57626664076770760413659262096, 4.28373735048288416703313255205, 4.89115725391632747879882280722, 5.76563673267363604751899689414, 6.56524989796469319329588569619, 7.20726031842460544111307676429, 7.922559735532104554314707682007