| L(s) = 1 | − 3·5-s + (−2 − 1.73i)7-s + 5.19i·11-s + 3.46i·13-s + 6·17-s − 1.73i·19-s − 5.19i·23-s + 4·25-s + 10.3i·29-s − 5.19i·31-s + (6 + 5.19i)35-s + 37-s + 3·41-s − 10·43-s − 6·47-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + (−0.755 − 0.654i)7-s + 1.56i·11-s + 0.960i·13-s + 1.45·17-s − 0.397i·19-s − 1.08i·23-s + 0.800·25-s + 1.92i·29-s − 0.933i·31-s + (1.01 + 0.878i)35-s + 0.164·37-s + 0.468·41-s − 1.52·43-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2818585877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2818585877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 5.19iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 5.19iT - 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 5.19iT - 71T^{2} \) |
| 73 | \( 1 + 3.46iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.291912165078873982046773746583, −7.60277070486980131044180321839, −6.99219211832104919156461737014, −6.54681280671259737121490475387, −5.12754387913780638659933960935, −4.40914325042995461081554599157, −3.77688240111925188310556803731, −2.94561479475134306526077932657, −1.54825719913142585626980472826, −0.10897078076687579224833851951,
1.01932724016355281992637817179, 2.88869269192788205398542592838, 3.35688742724583589401919427414, 4.02357223526393147613956281645, 5.48290297729960953092363597728, 5.72818608844261016504183430292, 6.75475957125653173449070368729, 7.81643054515517726610614443296, 8.091557451854792137476252105068, 8.782123655175865638387133269304
