| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1931 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1931 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.727962848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.727962848\) |
| \(L(1)\) |
\(\approx\) |
\(1.501336456\) |
| \(L(1)\) |
\(\approx\) |
\(1.501336456\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1931 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 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 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 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
−19.81478464041778028020604150323, −19.20257022937917396043443264913, −18.20100935491299242047789903855, −17.852069596572730975511681350946, −17.06850119218790645973428334394, −16.42963013333711559204589663658, −15.321903763793368180706158565828, −14.593860153206051735529393893143, −14.34890952748118754986779072896, −13.25156025305503121986130942848, −12.47063507351773485343842011026, −11.401477061655946984812619318622, −10.819947069789041551350757268243, −9.76822814414311926052147647687, −9.21587961107175725626248191205, −8.90725752115840614242171219187, −7.73631589762865542198888194041, −7.27786181533193828554362079939, −6.418020307952072325972565068365, −5.33788076508116768277015516672, −4.38581263932090415720127349232, −3.15335464293640083924355884389, −2.28336066303571707970173301523, −1.714601307885541751469015559528, −0.918443109348657983819643774463,
0.918443109348657983819643774463, 1.714601307885541751469015559528, 2.28336066303571707970173301523, 3.15335464293640083924355884389, 4.38581263932090415720127349232, 5.33788076508116768277015516672, 6.418020307952072325972565068365, 7.27786181533193828554362079939, 7.73631589762865542198888194041, 8.90725752115840614242171219187, 9.21587961107175725626248191205, 9.76822814414311926052147647687, 10.819947069789041551350757268243, 11.401477061655946984812619318622, 12.47063507351773485343842011026, 13.25156025305503121986130942848, 14.34890952748118754986779072896, 14.593860153206051735529393893143, 15.321903763793368180706158565828, 16.42963013333711559204589663658, 17.06850119218790645973428334394, 17.852069596572730975511681350946, 18.20100935491299242047789903855, 19.20257022937917396043443264913, 19.81478464041778028020604150323
