| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 19-s − 21-s + 23-s + 25-s + 27-s + 29-s + 31-s − 33-s + 35-s + 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s + 51-s + 55-s + 57-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 19-s − 21-s + 23-s + 25-s + 27-s + 29-s + 31-s − 33-s + 35-s + 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s + 51-s + 55-s + 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.126906898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126906898\) |
| \(L(1)\) |
\(\approx\) |
\(1.294592815\) |
| \(L(1)\) |
\(\approx\) |
\(1.294592815\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 53 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.390663462306605590031068774570, −25.62608894054707535733870315909, −24.73246118714555409950848693577, −23.4388269693920718012640105169, −23.01040643286516072997691409734, −21.535436083192605593337740889217, −20.60693619532594576256785494368, −19.82740498422471832989044028388, −18.89298562309205407819976190181, −18.38167651869668003808851396485, −16.451811720935146498179950645840, −15.79118775834226500206720231731, −15.090825317729664152619400229993, −13.75905084664617123156775030985, −12.99969416797668501157150010651, −11.97311222866284347670039451412, −10.57244473138611978037365967536, −9.602436850505155260130549297122, −8.41125982231489015252954310901, −7.68126416406476464507205425000, −6.55847484596472293794186017645, −4.87782714922705169058444728020, −3.43910389946598120108000679950, −2.97908316809539758440028916145, −0.93571552954413905515698748295,
0.93571552954413905515698748295, 2.97908316809539758440028916145, 3.43910389946598120108000679950, 4.87782714922705169058444728020, 6.55847484596472293794186017645, 7.68126416406476464507205425000, 8.41125982231489015252954310901, 9.602436850505155260130549297122, 10.57244473138611978037365967536, 11.97311222866284347670039451412, 12.99969416797668501157150010651, 13.75905084664617123156775030985, 15.090825317729664152619400229993, 15.79118775834226500206720231731, 16.451811720935146498179950645840, 18.38167651869668003808851396485, 18.89298562309205407819976190181, 19.82740498422471832989044028388, 20.60693619532594576256785494368, 21.535436083192605593337740889217, 23.01040643286516072997691409734, 23.4388269693920718012640105169, 24.73246118714555409950848693577, 25.62608894054707535733870315909, 26.390663462306605590031068774570
