| L(s) = 1 | + 2i·5-s + 2·7-s + 4i·13-s + 2·17-s + 4i·19-s + 4·23-s + 25-s − 6i·29-s − 2·31-s + 4i·35-s − 8i·37-s − 2·41-s − 4i·43-s − 12·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s + 0.755·7-s + 1.10i·13-s + 0.485·17-s + 0.917i·19-s + 0.834·23-s + 0.200·25-s − 1.11i·29-s − 0.359·31-s + 0.676i·35-s − 1.31i·37-s − 0.312·41-s − 0.609i·43-s − 1.75·47-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26215 + 0.522801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26215 + 0.522801i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.70501075944221256709369129947, −11.08533281834066889403621622620, −10.15853051390193707543673703059, −9.120211203359284962717171211114, −7.974550773834651301841793675608, −7.08223520525305021404437283726, −6.05781198711666120381656090210, −4.74473605100646981953288039781, −3.46003353815281816574250306079, −1.92438765299020886180452640903,
1.21356797142999595033969437009, 3.11283226319301725463295457547, 4.78567668159838626838443620956, 5.32602307359420780557817672875, 6.85596935270793405638480266063, 8.052530348861869333432227811207, 8.683717132121082251508571023729, 9.770654386271759173857043855634, 10.85022435549390451281088218098, 11.66806025337467352379219880264
