| L(s) = 1 | + 3.55·5-s + 1.45·7-s − 2.38·11-s − 0.979·13-s + 5.74·17-s + 2.24·19-s − 4.53·23-s + 7.63·25-s − 0.145·29-s − 0.733·31-s + 5.16·35-s + 37-s + 7.70·41-s + 8.41·43-s + 4.79·47-s − 4.88·49-s + 10.8·53-s − 8.49·55-s − 5.25·59-s + 2·61-s − 3.48·65-s + 0.0594·67-s + 4.79·71-s − 6.56·73-s − 3.47·77-s − 9.22·79-s + 11.5·83-s + ⋯ |
| L(s) = 1 | + 1.58·5-s + 0.549·7-s − 0.720·11-s − 0.271·13-s + 1.39·17-s + 0.515·19-s − 0.945·23-s + 1.52·25-s − 0.0269·29-s − 0.131·31-s + 0.873·35-s + 0.164·37-s + 1.20·41-s + 1.28·43-s + 0.699·47-s − 0.698·49-s + 1.49·53-s − 1.14·55-s − 0.684·59-s + 0.256·61-s − 0.432·65-s + 0.00726·67-s + 0.568·71-s − 0.768·73-s − 0.395·77-s − 1.03·79-s + 1.26·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.698815734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.698815734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 - 3.55T + 5T^{2} \) |
| 7 | \( 1 - 1.45T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 + 0.979T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 4.53T + 23T^{2} \) |
| 29 | \( 1 + 0.145T + 29T^{2} \) |
| 31 | \( 1 + 0.733T + 31T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 - 8.41T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 0.0594T + 67T^{2} \) |
| 71 | \( 1 - 4.79T + 71T^{2} \) |
| 73 | \( 1 + 6.56T + 73T^{2} \) |
| 79 | \( 1 + 9.22T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 - 7.95T + 97T^{2} \) |
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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
−9.045728748405683463810553244595, −7.954975476176018936603238034718, −7.50127697769584691249975030237, −6.37265307012552705402664473818, −5.56590360196720040574011649752, −5.31831124838661156647411654842, −4.15197149228899984149627423272, −2.87275015259803319761765672185, −2.13168239314445263726884620584, −1.10588166152722100365828003701,
1.10588166152722100365828003701, 2.13168239314445263726884620584, 2.87275015259803319761765672185, 4.15197149228899984149627423272, 5.31831124838661156647411654842, 5.56590360196720040574011649752, 6.37265307012552705402664473818, 7.50127697769584691249975030237, 7.954975476176018936603238034718, 9.045728748405683463810553244595
